Library UniMath.Bicategories.DisplayedBicats.FiberBicategory.FiberBicategory1

Bicategories

Benedikt Ahrens, Marco Maggesi February 2018 *********************************************************************************

Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.DisplayedCats.Constructions.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Unitors.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat.
Require Import UniMath.Bicategories.DisplayedBicats.FiberCategory.

Local Open Scope cat.
Local Open Scope mor_disp_scope.

Section Strict_Fiber_Bicat.
  Context {C : bicat}
          (D : disp_prebicat C)
          (h : local_iso_cleaving D)
          (c : C).

  Definition strict_fiber_bicat_1_id_comp_cells : prebicat_1_id_comp_cells.
  Show proof.
    exists (discrete_fiber_precategory_data D h c).
    red. cbn. intros d d' f f'.
    exact (f ==>[id2 (identity c)] f').

  Definition strict_fiber_bicat_data : prebicat_data.
  Show proof.
    exists strict_fiber_bicat_1_id_comp_cells.
    repeat split; cbn.
    - intros. exact (disp_id2 _).
    - intros d d' ff.
      set (PP := disp_local_iso_cleaving_invertible_2cell h (id_disp d;; ff) (idempunitor c)).
      set (RR := PP •• disp_lunitor ff).
      assert (Heq : (idempunitor c) • lunitor (identity c) = id2 (identity c)).
      { abstract (apply linvunitor_lunitor). }
      exact (transportf (λ x, _ ==>[x] _) Heq RR).
    - intros d d' ff.
      assert (Heq : (idempunitor c) • runitor (identity c) = id2 (identity c)).
      { abstract (cbn
                  ; rewrite <- lunitor_runitor_identity, linvunitor_lunitor
                  ; reflexivity).
      }
      set (PP := disp_local_iso_cleaving_invertible_2cell h (ff;; id_disp d') (idempunitor c)).
      exact (transportf (λ x, _ ==>[x] _) Heq (PP •• disp_runitor ff)).
    - intros d d' ff.
      set (PP := disp_inv_cell
                   (disp_local_iso_cleaving_invertible_2cell h (id_disp d;; ff) (idempunitor c))).
      assert (Heq : linvunitor (identity c) • (idempunitor c)^-1 = id2 (identity c)).
      { abstract (apply linvunitor_lunitor). }
      exact (transportf (λ x, _ ==>[x] _) Heq (disp_linvunitor ff •• PP)).
    - cbn. intros d d' ff.
      set (PP := disp_inv_cell
                   (disp_local_iso_cleaving_invertible_2cell h (ff;; id_disp d') (idempunitor c))).
      assert (Heq : rinvunitor (identity c) • (idempunitor c)^-1 = id2 (identity c)).
      { unfold idempunitor. cbn.
        abstract (rewrite lunitor_runitor_identity, rinvunitor_runitor
                  ; reflexivity).
      }
      exact (transportf (λ x, _ ==>[x] _) Heq (disp_rinvunitor ff •• PP)).
    - intros d0 d1 d2 d3 ff gg hh.
      assert (((idempunitor c) • ((idempunitor c) ▹ identity c)) • rassociator _ _ _ • ((identity c ◃ lunitor _) • lunitor _) = id2 _) as Heq.
      {
        abstract
          (cbn ;
           rewrite !lwhisker_hcomp, !rwhisker_hcomp ;
           rewrite lunitor_V_id_is_left_unit_V_id ;
           rewrite !vassocl ;
           rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)) ;
           rewrite <- triangle_l_inv ;
           rewrite !vassocl ;
           rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)) ;
           rewrite lassociator_rassociator, id2_left ;
           rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
           rewrite <- hcomp_vcomp ;
           rewrite id2_left, linvunitor_lunitor ;
           rewrite hcomp_identity, id2_left ;
           rewrite <- lunitor_V_id_is_left_unit_V_id ;
           rewrite linvunitor_lunitor ;
           reflexivity
          ).
      }
      refine (transportf (λ z, _ ==>[ z ] _) Heq _).
      cbn.
      refine (_ •• disp_rassociator ff gg hh •• _).
      + refine (disp_local_iso_cleaving_invertible_2cell h (local_iso_cleaving_1cell h (ff;; gg) (idempunitor c);; hh) (idempunitor c) •• _).
        refine (disp_rwhisker _ _).
        exact (disp_local_iso_cleaving_invertible_2cell h (ff ;; gg) (idempunitor c)).
      + refine (_ •• _).
        * refine (disp_lwhisker _ _).
          exact (disp_inv_cell (disp_local_iso_cleaving_invertible_2cell h (gg ;; hh) (idempunitor c))).
        * exact (disp_inv_cell ((disp_local_iso_cleaving_invertible_2cell h (ff;;local_iso_cleaving_1cell h (gg;; hh) (idempunitor c)) (idempunitor c)))).
    - intros d0 d1 d2 d3 ff gg hh.
      assert (((idempunitor c) • (identity c ◃ (idempunitor c))) • lassociator _ _ _ • ((lunitor _ ▹ identity c) • lunitor _) = id2 _) as Heq.
      {
        abstract
          (cbn ;
           rewrite !lwhisker_hcomp, !rwhisker_hcomp ;
           rewrite !vassocl ;
           rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)) ;
           rewrite triangle_r_inv ;
           rewrite !vassocl ;
           rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)) ;
           rewrite rassociator_lassociator, id2_left ;
           rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
           rewrite <- hcomp_vcomp ;
           rewrite <- lunitor_V_id_is_left_unit_V_id ;
           rewrite id2_left, linvunitor_lunitor ;
           rewrite hcomp_identity, id2_left ;
           apply linvunitor_lunitor
          ).
      }
      refine (transportf (λ z, _ ==>[ z ] _) Heq _).
      cbn.
      refine (_ •• disp_lassociator ff gg hh •• _).
      + refine (_ •• _).
        * exact (disp_local_iso_cleaving_invertible_2cell h (ff;;local_iso_cleaving_1cell h (gg;; hh) (idempunitor c)) (idempunitor c)).
        * refine (disp_lwhisker _ _).
          exact (disp_local_iso_cleaving_invertible_2cell h (gg ;; hh) (idempunitor c)).
      + refine (_ •• _).
        * refine (disp_rwhisker _ _).
          exact (disp_inv_cell (disp_local_iso_cleaving_invertible_2cell h (ff ;; gg) (idempunitor c))).
        * exact (disp_inv_cell (disp_local_iso_cleaving_invertible_2cell h (local_iso_cleaving_1cell h (ff;; gg) (idempunitor c);; hh) (idempunitor c))).
    - intros a b f1 f2 f3 x y.
      exact (transportf (λ z, _ ==>[ z ] _) (id2_left _) (x •• y)).
    - intros a1 a2 a3 f g1 g2 x.
      assert (linvunitor _ • (identity c ◃ id2 _) • lunitor _ = id2 (identity c)) as Heq.
      {
        abstract (rewrite lwhisker_id2 ;
                  rewrite id2_right ;
                  apply linvunitor_lunitor).
      }
      refine (transportf (λ z, _ ==>[ z ] _) Heq _).
      refine (_ •• (f ◃◃ x) •• _).
      + exact (disp_local_iso_cleaving_invertible_2cell h (f ;; g1) (idempunitor c)).
      + exact (disp_inv_cell (disp_local_iso_cleaving_invertible_2cell h (f ;; g2) (idempunitor c))).
    - intros a1 a2 a3 f1 f2 g x.
      assert (linvunitor _ • (id2 _ ▹ identity _) • lunitor _ = id2 (identity c)) as Heq.
      {
        abstract (rewrite id2_rwhisker ;
                  rewrite id2_right ;
                  apply linvunitor_lunitor).
      }
      refine (transportf (λ z, _ ==>[ z ] _) Heq _).
      refine (_ •• (x ▹▹ g) •• _).
      + exact (disp_local_iso_cleaving_invertible_2cell h (f1 ;; g) (idempunitor c)).
      + exact (disp_inv_cell (disp_local_iso_cleaving_invertible_2cell h (f2 ;; g) (idempunitor c))).

  Local Arguments transportf {_} {_} {_} {_} {_} _.
  Local Arguments transportb {_} {_} {_} {_} {_} _.

  Definition strict_fiber_bicat_data_laws_vcomp_left
    : ∏ (a b : strict_fiber_bicat_data)
         (f g : strict_fiber_bicat_data ⟦ a, b ⟧)
         (x : f ==> g),
       id₂ f • x = x.
  Show proof.
    intros a b f g x ; cbn.
    rewrite disp_id2_left.
    rewrite transportfbinv.
    reflexivity.

  Definition strict_fiber_bicat_data_laws_vcomp_right
    : ∏ (a b : strict_fiber_bicat_data)
         (f g : strict_fiber_bicat_data ⟦ a, b ⟧)
         (x : f ==> g),
       x • id₂ g = x.
  Show proof.
    intros a b f g x ; cbn.
    rewrite disp_id2_right.
    unfold transportb.
    rewrite transport_f_f.
    rewrite transportf_set.
    - reflexivity.
    - apply C.

  Definition strict_fiber_bicat_data_laws_vcomp_assoc
    : ∏ (a b : strict_fiber_bicat_data)
        (f₁ f₂ f₃ f₄ : strict_fiber_bicat_data ⟦ a, b ⟧)
        (x : f₁ ==> f₂)
        (y : f₂ ==> f₃)
        (z : f₃ ==> f₄),
      x • (y • z) = (x • y) • z.
  Show proof.
    intros a b f₁ f₂ f₃ f₄ x y z ; cbn.
    rewrite disp_mor_transportf_prewhisker.
    rewrite disp_vassocr.
    rewrite disp_mor_transportf_postwhisker.
    unfold transportb.
    rewrite !transport_f_f.
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, f₁ ==>[ α] f₄)).
    apply C.

  Definition strict_fiber_bicat_data_laws_rwhisker_id2
    : ∏ (a₁ a₂ a₃ : strict_fiber_bicat_data)
        (f : strict_fiber_bicat_data ⟦ a₁ , a₂ ⟧)
        (g : strict_fiber_bicat_data ⟦ a₂, a₃ ⟧),
      f ◃ id₂ g = id₂ (f · g).
  Show proof.
    intros a₁ a₂ a₃ f g ; cbn.
    rewrite disp_lwhisker_id2.
    unfold transportb.
    rewrite disp_mor_transportf_prewhisker.
    rewrite disp_mor_transportf_postwhisker.
    rewrite transport_f_f.
    rewrite disp_id2_right.
    unfold transportb.
    rewrite disp_mor_transportf_postwhisker.
    rewrite transport_f_f.
    apply (@transportf_transpose_left _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    refine (disp_vcomp_rinv
              (disp_local_iso_cleaving_invertible_2cell h (f;; g) (idempunitor c))
              @ _).
    unfold transportb.
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply C.

  Definition strict_fiber_bicat_data_laws_id2_lwhisker
    : ∏ (a₁ a₂ a₃ : strict_fiber_bicat_data)
        (f : strict_fiber_bicat_data ⟦ a₁ , a₂ ⟧)
        (g : strict_fiber_bicat_data ⟦ a₂ , a₃ ⟧),
      id₂ f ▹ g = id₂ (f · g).
  Show proof.
    intros a₁ a₂ a₃ f g ; cbn.
    rewrite disp_id2_rwhisker.
    unfold transportb.
    rewrite disp_mor_transportf_prewhisker.
    rewrite disp_mor_transportf_postwhisker.
    rewrite transport_f_f.
    rewrite disp_id2_right.
    unfold transportb.
    rewrite disp_mor_transportf_postwhisker.
    rewrite transport_f_f.
    apply (@transportf_transpose_left _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    refine (disp_vcomp_rinv
              (disp_local_iso_cleaving_invertible_2cell h (f;; g) (idempunitor c))
              @ _).
    unfold transportb.
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply C.

  Definition strict_fiber_bicat_data_laws_vcomp_lwhisker
    : ∏ (a₁ a₂ a₃ : strict_fiber_bicat_data)
         (f : strict_fiber_bicat_data ⟦ a₁ , a₂ ⟧)
         (g₁ g₂ g₃ : strict_fiber_bicat_data ⟦ a₂ , a₃ ⟧)
         (x : g₁ ==> g₂)
         (y : g₂ ==> g₃),
       (f ◃ x) • (f ◃ y) = f ◃ (x • y).
  Show proof.
    intros a₁ a₂ a₃ f g₁ g₂ g₃ x y ; cbn.
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_prewhisker.
      }
      apply maponpaths.
      apply disp_mor_transportf_postwhisker.
    }
    do 2 refine (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)
                                _ _ _ _ _ _ @ _).
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              refine (disp_vassocr _ _ _ @ _).
              apply maponpaths.
              etrans.
              {
                apply maponpaths_2.
                exact (disp_vcomp_linv
                         (disp_local_iso_cleaving_invertible_2cell
                            h (f;; g₂)
                            (idempunitor c))).
              }
              etrans.
              {
                apply disp_mor_transportf_postwhisker.
              }
              etrans.
              {
                apply maponpaths.
                apply disp_id2_left.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            etrans.
            {
              apply maponpaths_2.
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              apply disp_lwhisker_vcomp.
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths_2.
        etrans.
        {
          apply maponpaths.
          apply disp_rwhisker_transport_right.
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply disp_mor_transportf_postwhisker.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      apply disp_vassocl.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply C.

  Definition strict_fiber_bicat_data_laws_vcomp_rwhisker
    : ∏ (a₁ a₂ a₃ : strict_fiber_bicat_data)
        (f₁ f₂ f₃ : strict_fiber_bicat_data ⟦ a₁ , a₂ ⟧)
        (g : strict_fiber_bicat_data ⟦ a₂ , a₃ ⟧)
        (x : f₁ ==> f₂)
        (y : f₂ ==> f₃),
      (x ▹ g) • (y ▹ g) = (x • y) ▹ g.
  Show proof.
    intros a₁ a₂ a₃ f₁ f₂ f₃ f₄ x y ; cbn.
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_postwhisker.
      }
      apply maponpaths.
      apply disp_mor_transportf_prewhisker.
    }
    do 2 refine (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)
                                _ _ _ _ _ _ @ _).
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              refine (disp_vassocr _ _ _ @ _).
              apply maponpaths.
              etrans.
              {
                apply maponpaths_2.
                exact (disp_vcomp_linv
                         (disp_local_iso_cleaving_invertible_2cell
                            h (f₂;; f₄) (idempunitor c))).
              }
              etrans.
              {
                apply disp_mor_transportf_postwhisker.
              }
              etrans.
              {
                apply maponpaths.
                apply disp_id2_left.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            etrans.
            {
              apply maponpaths_2.
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocr _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          refine (disp_vassocl _ _ _ @ _).
          apply maponpaths.
          etrans.
          {
            apply maponpaths.
            apply disp_rwhisker_vcomp.
          }
          apply disp_mor_transportf_prewhisker.
        }
        etrans.
        {
          apply maponpaths_2.
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths_2.
        etrans.
        {
          apply maponpaths.
          apply disp_rwhisker_transport_left_new.
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply disp_mor_transportf_postwhisker.
    }
    refine (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)
                           _ _ _ _ _ _ @ _).
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply C.

  Definition strict_fiber_bicat_data_laws_vcomp_lunitor
    : ∏ (a b : strict_fiber_bicat_data)
        (f g : strict_fiber_bicat_data ⟦ a, b ⟧)
        (x : f ==> g),
      (id₁ a ◃ x) • lunitor g = lunitor f • x.
  Show proof.
    intros a b f g x ; cbn.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    rewrite !disp_mor_transportf_postwhisker.
    rewrite !transport_f_f.
    etrans.
    {
      apply maponpaths.
      rewrite disp_vassocl.
      do 2 apply maponpaths.
      rewrite disp_vassocr.
      apply maponpaths.
      apply maponpaths_2.
      apply (disp_vcomp_linv (disp_local_iso_cleaving_invertible_2cell
                                h (id_disp a;; g) (idempunitor c))).
    }
    unfold transportb.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      do 2 apply maponpaths.
      apply disp_id2_left.
    }
    unfold transportb.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      apply maponpaths.
      rewrite disp_vassocl.
      do 2 apply maponpaths.
      apply disp_vcomp_lunitor.
    }
    unfold transportb.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    refine (_ @ _).
    {
      apply maponpaths.
      apply disp_vassocr.
    }
    unfold transportb.
    rewrite transport_f_f.
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply C.

  Definition strict_fiber_bicat_data_laws_vcomp_runitor
    : ∏ (a b : strict_fiber_bicat_data)
        (f g : strict_fiber_bicat_data ⟦ a, b ⟧)
        (x : f ==> g),
      (x ▹ id₁ b) • runitor g = runitor f • x.
  Show proof.
    intros a b f g x ; cbn.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    rewrite !disp_mor_transportf_postwhisker.
    rewrite !transport_f_f.
    etrans.
    {
      apply maponpaths.
      rewrite disp_vassocl.
      do 2 apply maponpaths.
      rewrite disp_vassocr.
      apply maponpaths.
      apply maponpaths_2.
      apply (disp_vcomp_linv (disp_local_iso_cleaving_invertible_2cell
                                h (g;;id_disp b) (idempunitor c))).
    }
    unfold transportb.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      do 2 apply maponpaths.
      apply disp_id2_left.
    }
    unfold transportb.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      apply maponpaths.
      rewrite disp_vassocl.
      do 2 apply maponpaths.
      apply disp_vcomp_runitor.
    }
    unfold transportb.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    refine (_ @ _).
    {
      apply maponpaths.
      apply disp_vassocr.
    }
    unfold transportb.
    rewrite transport_f_f.
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply C.

  Definition strict_fiber_bicat_data_laws_lwhisker_lwhisker
    : ∏ (a₁ a₂ a₃ a₄ : strict_fiber_bicat_data)
        (f₁ : strict_fiber_bicat_data ⟦ a₁ , a₂ ⟧)
        (f₂ : strict_fiber_bicat_data ⟦ a₂ , a₃ ⟧)
        (f₃ f₄ : strict_fiber_bicat_data ⟦ a₃ , a₄ ⟧)
        (x : f₃ ==> f₄),
      (f₁ ◃ (f₂ ◃ x)) • lassociator f₁ f₂ f₄
      =
      lassociator f₁ f₂ f₃ • (f₁ · f₂ ◃ x).
  Show proof.
    intros a₁ a₂ a₃ a₄ f₁ f₂ g₁ g₂ x ; cbn.
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_postwhisker.
      }
      etrans.
      {
        apply maponpaths.
        apply disp_mor_transportf_prewhisker.
      }
      do 2 apply maponpaths.
      etrans.
      {
        apply maponpaths_2.
        etrans.
        {
          apply maponpaths_2.
          etrans.
          {
            apply maponpaths.
            apply disp_rwhisker_transport_right.
          }
          apply disp_mor_transportf_prewhisker.
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply disp_mor_transportf_postwhisker.
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_postwhisker.
      }
      apply maponpaths.
      apply disp_mor_transportf_prewhisker.
    }
    refine (!_).
    do 3 (refine (@transport_f_f _ (λ z : id₁ c ==> id₁ c, _ ==>[ z ] _)
                                   _ _ _ _ _ _ @ _)).
    do 2 refine (_ @ !(@transport_f_f _ (λ z : id₁ c ==> id₁ c, _ ==>[ z ] _)
                                        _ _ _ _ _ _)).
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            refine (disp_vassocr _ _ _ @ _).
            etrans.
            {
              apply maponpaths.
              etrans.
              {
                apply maponpaths_2.
                etrans.
                {
                  apply maponpaths_2.
                  refine (disp_vassocr _ _ _ @ _).
                  etrans.
                  {
                    apply maponpaths.
                    etrans.
                    {
                      apply maponpaths_2.
                      refine (disp_vassocr _ _ _ @ _).
                      etrans.
                      {
                        apply maponpaths.
                        etrans.
                        {
                          apply maponpaths_2.
                          exact (disp_vcomp_linv
                                   (disp_local_iso_cleaving_invertible_2cell
                                      h
                                      (f₁;; local_iso_cleaving_1cell
                                         h (f₂;; g₂) (idempunitor c))
                                      (idempunitor c))).
                        }
                        etrans.
                        {
                          apply disp_mor_transportf_postwhisker.
                        }
                        etrans.
                        {
                          apply maponpaths.
                          apply disp_id2_left.
                        }
                        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                      }
                      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                    }
                    apply disp_mor_transportf_postwhisker.
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_postwhisker.
              }
              apply disp_mor_transportf_postwhisker.
            }
            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              refine (disp_vassocr _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths_2.
                  refine (disp_vassocr _ _ _ @ _).
                  etrans.
                  {
                    apply maponpaths.
                    etrans.
                    {
                      apply maponpaths_2.
                      etrans.
                      {
                        apply disp_lwhisker_vcomp.
                      }
                      apply maponpaths.
                      etrans.
                      {
                        apply maponpaths.
                        apply disp_vassocl.
                      }
                      etrans.
                      {
                        apply disp_rwhisker_transport_right.
                      }
                      etrans.
                      {
                        apply maponpaths.
                        etrans.
                        {
                          apply maponpaths.
                          etrans.
                          {
                            apply maponpaths.
                            exact (disp_vcomp_linv
                                     (disp_local_iso_cleaving_invertible_2cell
                                        h (f₂;; g₂) (idempunitor c))).
                          }
                          etrans.
                          {
                            apply disp_mor_transportf_prewhisker.
                          }
                          etrans.
                          {
                            apply maponpaths.
                            apply disp_id2_right.
                          }
                          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                        }
                        apply disp_rwhisker_transport_right.
                      }
                      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                    }
                    etrans.
                    {
                      apply maponpaths_2.
                      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                    }
                    apply disp_mor_transportf_postwhisker.
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_postwhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          etrans.
          {
            apply maponpaths_2.
            etrans.
            {
              apply maponpaths_2.
              apply disp_lwhisker_vcomp_alt.
            }
            etrans.
            {
              apply disp_mor_transportf_postwhisker.
            }
            etrans.
            {
              apply maponpaths.
              refine (disp_vassocl _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths.
                  apply disp_lwhisker_lwhisker.
                }
                apply disp_mor_transportf_prewhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
          }
          apply disp_mor_transportf_postwhisker.
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply disp_mor_transportf_prewhisker.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          refine (disp_vassocl _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths.
              refine (disp_vassocl _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths.
                  apply disp_vcomp_whisker_alt.
                }
                apply disp_mor_transportf_prewhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_prewhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply disp_mor_transportf_prewhisker.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        refine (disp_vassocl _ _ _ @ _).
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocl _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths.
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths.
                  refine (disp_vassocl _ _ _ @ _).
                  etrans.
                  {
                    apply maponpaths.
                    etrans.
                    {
                      apply maponpaths.
                      refine (disp_vassocr _ _ _ @ _).
                      etrans.
                      {
                        apply maponpaths.
                        etrans.
                        {
                          apply maponpaths_2.
                          refine (disp_vassocr _ _ _ @ _).
                          etrans.
                          {
                            apply maponpaths.
                            etrans.
                            {
                              apply maponpaths_2.
                              exact (disp_vcomp_linv
                                       (disp_local_iso_cleaving_invertible_2cell
                                          h
                                          (local_iso_cleaving_1cell
                                             h (f₁;; f₂) (idempunitor c);; g₁)
                                          (idempunitor c))).
                            }
                            etrans.
                            {
                              apply disp_mor_transportf_postwhisker.
                            }
                            etrans.
                            {
                              apply maponpaths.
                              apply disp_id2_left.
                            }
                            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                          }
                          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                        }
                        apply disp_mor_transportf_postwhisker.
                      }
                      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                    }
                    apply disp_mor_transportf_prewhisker.
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_prewhisker.
              }
              apply disp_mor_transportf_prewhisker.
            }
            apply disp_mor_transportf_prewhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths.
              apply disp_vassocr.
            }
            apply disp_mor_transportf_prewhisker.
          }
          apply disp_mor_transportf_prewhisker.
        }
        apply disp_mor_transportf_prewhisker.
      }
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths.
            apply disp_vassocr.
          }
          apply disp_mor_transportf_prewhisker.
        }
        apply disp_mor_transportf_prewhisker.
      }
      etrans.
      {
        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
      }
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        apply disp_vassocr.
      }
      apply disp_mor_transportf_prewhisker.
    }
    do 2 refine (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)
                                _ _ _ _ _ _ @ _).
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply cellset_property.

  Definition strict_fiber_bicat_data_laws_rwhisker_lwhisker
    : ∏ (a₁ a₂ a₃ a₄ : strict_fiber_bicat_data)
        (f₁ : strict_fiber_bicat_data ⟦ a₁ , a₂ ⟧)
        (f₂ f₃ : strict_fiber_bicat_data ⟦ a₂ , a₃ ⟧)
        (f₄ : strict_fiber_bicat_data ⟦ a₃ , a₄ ⟧)
        (x : f₂ ==> f₃),
      (f₁ ◃ (x ▹ f₄)) • lassociator f₁ f₃ f₄
      =
      lassociator f₁ f₂ f₄ • ((f₁ ◃ x) ▹ f₄).
  Show proof.
    intros a₁ a₂ a₃ a₄ f₁ f₂ f₃ f₄ x ; cbn.
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_postwhisker.
      }
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_prewhisker.
      }
      apply maponpaths.
      etrans.
      {
        apply maponpaths_2.
        etrans.
        {
          apply maponpaths_2.
          etrans.
          {
            apply maponpaths.
            apply disp_rwhisker_transport_right.
          }
          apply disp_mor_transportf_prewhisker.
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply disp_mor_transportf_postwhisker.
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_postwhisker.
      }
      apply maponpaths.
      apply disp_mor_transportf_prewhisker.
    }
    refine (!_).
    do 3 (refine (@transport_f_f _ (λ z : id₁ c ==> id₁ c, _ ==>[ z ] _)
                                   _ _ _ _ _ _ @ _)).
    do 2 refine (_ @ !(@transport_f_f _ (λ z : id₁ c ==> id₁ c, _ ==>[ z ] _)
                                      _ _ _ _ _ _)).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        refine (disp_vassocl _ _ _ @ _).
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              refine (disp_vassocr _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths_2.
                  refine (disp_vassocr _ _ _ @ _).
                  etrans.
                  {
                    apply maponpaths.
                    etrans.
                    {
                      apply maponpaths_2.
                      exact (disp_vcomp_linv
                               (disp_local_iso_cleaving_invertible_2cell
                                  h
                                  (f₁;; local_iso_cleaving_1cell
                                     h (f₃;; f₄) (idempunitor c))
                                  (idempunitor c))).
                    }
                    etrans.
                    {
                      apply disp_mor_transportf_postwhisker.
                    }
                    etrans.
                    {
                      apply maponpaths.
                      apply disp_id2_left.
                    }
                    apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_postwhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            refine (disp_vassocr _ _ _ @ _).
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              etrans.
              {
                apply maponpaths_2.
                refine (disp_vassocr _ _ _ @ _).
                etrans.
                {
                  apply maponpaths.
                  etrans.
                  {
                    apply maponpaths_2.
                    refine (disp_lwhisker_vcomp _ _ @ _).
                    etrans.
                    {
                      apply maponpaths.
                      etrans.
                      {
                        apply maponpaths.
                        refine (disp_vassocl _ _ _ @ _).
                        etrans.
                        {
                          apply maponpaths.
                          etrans.
                          {
                            apply maponpaths.
                            exact (disp_vcomp_linv
                                     (disp_local_iso_cleaving_invertible_2cell
                                        h (f₃;; f₄) (idempunitor c))).
                          }
                          apply disp_mor_transportf_prewhisker.
                        }
                        etrans.
                        {
                          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                        }
                        etrans.
                        {
                          apply maponpaths.
                          apply disp_id2_right.
                        }
                        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                      }
                      apply disp_rwhisker_transport_right.
                    }
                    apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                  }
                  apply disp_mor_transportf_postwhisker.
                }
                apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
              }
              apply disp_mor_transportf_postwhisker.
            }
            apply disp_mor_transportf_postwhisker.
          }
          etrans.
          {
            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }


    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          etrans.
          {
            apply maponpaths_2.
            etrans.
            {
              apply maponpaths_2.
              apply disp_lwhisker_vcomp_alt.
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply disp_mor_transportf_postwhisker.
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply disp_mor_transportf_prewhisker.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          etrans.
          {
            apply maponpaths_2.
            refine (disp_vassocl _ _ _ @ _).
            do 2 apply maponpaths.
            apply disp_rwhisker_lwhisker.
          }
          unfold transportb.
          apply disp_mor_transportf_postwhisker.
        }
        etrans.
        {
          apply disp_mor_transportf_postwhisker.
        }
        apply maponpaths.
        do 2 apply maponpaths_2.
        apply disp_mor_transportf_prewhisker.
      }
      etrans.
      {
        apply disp_mor_transportf_prewhisker.
      }
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          apply disp_mor_transportf_postwhisker.
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply disp_mor_transportf_prewhisker.
    }
    do 2 refine (@transport_f_f _ (λ z : id₁ c ==> id₁ c, _ ==>[ z ] _)
                                _ _ _ _ _ _ @ _).
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          etrans.
          {
            apply maponpaths.
            apply disp_rwhisker_transport_left_new.
          }
          apply disp_mor_transportf_prewhisker.
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply disp_mor_transportf_prewhisker.
    }
    refine (@transport_f_f _ (λ z : id₁ c ==> id₁ c, _ ==>[ z ] _)
                           _ _ _ _ _ _ @ _).
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocr _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          refine (disp_vassocl _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths.
              refine (disp_vassocl _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths.
                  refine (disp_vassocr _ _ _ @ _).
                  etrans.
                  {
                    apply maponpaths.
                    etrans.
                    {
                      apply maponpaths_2.
                      exact (disp_vcomp_linv
                               (disp_local_iso_cleaving_invertible_2cell
                                  h
                                  (local_iso_cleaving_1cell
                                     h (f₁;; f₂) (idempunitor c);; f₄) (idempunitor c))).
                    }
                    etrans.
                    {
                      apply disp_mor_transportf_postwhisker.
                    }
                    etrans.
                    {
                      apply maponpaths.
                      apply disp_id2_left.
                    }
                    apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_prewhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_prewhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        refine (disp_vassocl _ _ _ @ _).
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocl _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths.
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths.
                  etrans.
                  {
                    apply maponpaths_2.
                    apply disp_rwhisker_vcomp.
                  }
                  etrans.
                  {
                    apply disp_mor_transportf_postwhisker.
                  }
                  apply maponpaths.
                  etrans.
                  {
                    apply maponpaths_2.
                    etrans.
                    {
                      apply maponpaths.
                      refine (disp_vassocr _ _ _ @ _).
                      etrans.
                      {
                        apply maponpaths.
                        etrans.
                        {
                          apply maponpaths_2.
                          refine (disp_vassocr _ _ _ @ _).
                          etrans.
                          {
                            apply maponpaths.
                            etrans.
                            {
                              apply maponpaths_2.
                              exact (disp_vcomp_linv
                                       (disp_local_iso_cleaving_invertible_2cell
                                          h (f₁;; f₂) (idempunitor c))).
                            }
                            etrans.
                            {
                              apply disp_mor_transportf_postwhisker.
                            }
                            etrans.
                            {
                              apply maponpaths.
                              apply disp_id2_left.
                            }
                            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                          }
                          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                        }
                        apply disp_mor_transportf_postwhisker.
                      }
                      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                    }
                    etrans.
                    {
                      apply disp_rwhisker_transport_left_new.
                    }
                    etrans.
                    {
                      apply maponpaths.
                      apply disp_rwhisker_vcomp_alt.
                    }
                    apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                  }
                  apply disp_mor_transportf_postwhisker.
                }
                etrans.
                {
                  apply maponpaths.
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_prewhisker.
              }
              apply disp_mor_transportf_prewhisker.
            }
            apply disp_mor_transportf_prewhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        refine (disp_vassocl _ _ _ @ _).
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocl _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths.
              refine (disp_vassocl _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths.
                  apply disp_vassocr.
                }
                apply disp_mor_transportf_prewhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_prewhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
      }
      apply disp_mor_transportf_prewhisker.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply cellset_property.

  Definition strict_fiber_bicat_data_laws_rwhisker_rwhisker
    : ∏ (a₁ a₂ a₃ a₄ : strict_fiber_bicat_data)
         (f₁ f₂ : strict_fiber_bicat_data ⟦ a₁ , a₂ ⟧)
         (f₃ : strict_fiber_bicat_data ⟦ a₂ , a₃ ⟧)
         (f₄ : strict_fiber_bicat_data ⟦ a₃ , a₄ ⟧)
         (x : f₁ ==> f₂),
       lassociator f₁ f₃ f₄ • ((x ▹ f₃) ▹ f₄)
       =
       (x ▹ f₃ · f₄) • lassociator f₂ f₃ f₄.
  Show proof.
    intros a₁ a₂ a₃ a₄ f₁ f₂ f₃ f₄ x ; cbn.
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_postwhisker.
      }
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_prewhisker.
      }
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          etrans.
          {
            apply maponpaths.
            apply disp_rwhisker_transport_left_new.
          }
          apply disp_mor_transportf_prewhisker.
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply disp_mor_transportf_prewhisker.
    }
    do 3 refine (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)
                                _ _ _ _ _ _ @ _).
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_postwhisker.
      }
      apply maponpaths.
      apply disp_mor_transportf_prewhisker.
    }
    do 2 refine (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)
                                _ _ _ _ _ _ @ _).
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        refine (disp_vassocr _ _ _ @ _).
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths_2.
            refine (disp_vassocr _ _ _ @ _).
            etrans.
            {
              apply maponpaths.
              etrans.
              {
                apply maponpaths_2.
                refine (disp_vassocr _ _ _ @ _).
                etrans.
                {
                  apply maponpaths.
                  etrans.
                  {
                    apply maponpaths_2.
                    exact (disp_vcomp_linv
                             (disp_local_iso_cleaving_invertible_2cell
                                h
                                (f₂;; local_iso_cleaving_1cell h (f₃;; f₄) (idempunitor c))
                                (idempunitor c))).
                  }
                  apply disp_mor_transportf_postwhisker.
                }
                etrans.
                {
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply maponpaths.
                apply disp_id2_left.
              }
              etrans.
              {
                apply disp_mor_transportf_postwhisker.
              }
              etrans.
              {
                apply maponpaths.
                apply disp_mor_transportf_postwhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
          }
          apply disp_mor_transportf_postwhisker.
        }
        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
      }
      apply disp_mor_transportf_prewhisker.
    }
    do 2 refine (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)
                                _ _ _ _ _ _ @ _).
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        refine (disp_vassocl _ _ _ @ _).
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            refine (disp_vassocr _ _ _ @ _).
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              refine (disp_vassocl _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths.
                  refine (disp_vassocr _ _ _ @ _).
                  etrans.
                  {
                    apply maponpaths.
                    etrans.
                    {
                      apply maponpaths_2.
                      apply (disp_vcomp_linv
                               (disp_local_iso_cleaving_invertible_2cell
                                  h
                                  (local_iso_cleaving_1cell h (f₁;; f₃) (idempunitor c);; f₄)
                                  (idempunitor c))).
                    }
                    apply disp_mor_transportf_postwhisker.
                  }
                  etrans.
                  {
                    apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                  }
                  etrans.
                  {
                    apply maponpaths.
                    apply disp_id2_left.
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_prewhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              apply disp_rwhisker_vcomp.
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply disp_mor_transportf_prewhisker.
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths_2.
            etrans.
            {
              apply maponpaths.
              refine (disp_vassocr _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths_2.
                  refine (disp_vassocr _ _ _ @ _).
                  etrans.
                  {
                    apply maponpaths.
                    etrans.
                    {
                      apply maponpaths_2.
                      apply (disp_vcomp_linv
                               (disp_local_iso_cleaving_invertible_2cell
                                  h (f₁;; f₃)
                                  (idempunitor c))).
                    }
                    etrans.
                    {
                      apply disp_mor_transportf_postwhisker.
                    }
                    apply maponpaths.
                    apply disp_id2_left.
                  }
                  etrans.
                  {
                    apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_postwhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_rwhisker_transport_left_new.
          }
          apply disp_mor_transportf_postwhisker.
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply disp_mor_transportf_prewhisker.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths_2.
            apply disp_rwhisker_vcomp_alt.
          }
          apply disp_mor_transportf_postwhisker.
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply disp_mor_transportf_prewhisker.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        refine (disp_vassocr _ _ _ @ _).
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths_2.
            refine (disp_vassocr _ _ _ @ _).
            etrans.
            {
              apply maponpaths.
              etrans.
              {
                apply maponpaths_2.
                apply disp_rwhisker_rwhisker.
              }
              apply disp_mor_transportf_postwhisker.
            }
            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
          }
          apply disp_mor_transportf_postwhisker.
        }
        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
      }
      apply disp_mor_transportf_prewhisker.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              refine (disp_vassocr _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths_2.
                  refine (disp_vassocr _ _ _ @ _).
                  etrans.
                  {
                    apply maponpaths.
                    etrans.
                    {
                      apply maponpaths_2.
                      apply disp_vcomp_whisker_alt.
                    }
                    apply disp_mor_transportf_postwhisker.
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_postwhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            refine (disp_vassocr _ _ _ @ _).
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              etrans.
              {
                apply maponpaths_2.
                apply disp_vassocr.
              }
              apply disp_mor_transportf_postwhisker.
            }
            apply disp_mor_transportf_postwhisker.
          }
          etrans.
          {
            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply cellset_property.

  Definition strict_fiber_bicat_data_vcomp_whisker
    : ∏ (a₁ a₂ a₃ : strict_fiber_bicat_data)
         (f₁ f₂ : strict_fiber_bicat_data ⟦ a₁ , a₂ ⟧)
         (g₁ g₂ : strict_fiber_bicat_data ⟦ a₂ , a₃ ⟧)
         (x : f₁ ==> f₂) (y : g₁ ==> g₂),
       (x ▹ g₁) • (f₂ ◃ y) = (f₁ ◃ y) • (x ▹ g₂).
  Show proof.
    intros a₁ a₂ a₃ f₁ f₂ g₁ g₂ x y ; cbn.
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_prewhisker.
      }
      etrans.
      {
        apply maponpaths.
        apply disp_mor_transportf_postwhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_prewhisker.
      }
      etrans.
      {
        apply maponpaths.
        apply disp_mor_transportf_postwhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocr _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths_2.
          etrans.
          {
            refine (disp_vassocl _ _ _ @ _).
            apply maponpaths.
            etrans.
            {
              apply maponpaths.
              refine (disp_vassocr _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths_2.
                  exact (disp_vcomp_linv
                           (disp_local_iso_cleaving_invertible_2cell
                              h (f₁;; g₂) (idempunitor c))).
                }
                apply disp_mor_transportf_postwhisker.
              }
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths.
                  apply disp_id2_left.
                }
                apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_prewhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_postwhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        refine (disp_vassocl _ _ _ @ _).
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              refine (disp_vassocr _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths_2.
                  exact (disp_vcomp_linv
                           (disp_local_iso_cleaving_invertible_2cell
                              h (f₂;; g₁) (idempunitor c))).
                }
                apply disp_mor_transportf_postwhisker.
              }
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths.
                  apply disp_id2_left.
                }
                apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              apply disp_vcomp_whisker.
            }
            apply disp_mor_transportf_postwhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    refine (!_).
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        refine (disp_vassocl _ _ _ @ _).
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths.
            apply disp_vassocr.
          }
          apply disp_mor_transportf_prewhisker.
        }
        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    apply (@transportf_paths _ (λ α : id₁ c ==> id₁ c, _ ==>[ α] _)).
    apply cellset_property.

  Definition strict_fiber_bicat_data_laws_lunitor_linvunitor
    : ∏ (a b : strict_fiber_bicat_data)
         (f : strict_fiber_bicat_data ⟦ a, b ⟧),
       lunitor f • linvunitor f = id₂ (id₁ a · f).
  Show proof.
    intros a b f ; cbn.
    rewrite disp_mor_transportf_prewhisker.
    rewrite !transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite !transport_f_f.
    etrans.
    {
      apply maponpaths.
      rewrite disp_vassocl.
      do 2 apply maponpaths.
      rewrite disp_vassocr.
      apply maponpaths, maponpaths_2.
      apply disp_lunitor_linvunitor.
    }
    unfold transportb.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      do 2 apply maponpaths.
      apply disp_id2_left.
    }
    unfold transportb.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      apply maponpaths.
      apply (disp_vcomp_rinv (disp_local_iso_cleaving_invertible_2cell h (id_disp a;; f) (idempunitor c))).
    }
    unfold transportb.
    rewrite transport_f_f.
    apply (transportf_set (λ α : id₁ c ==> id₁ c, _ ==>[ α] _) _).
    apply C.

  Definition strict_fiber_bicat_data_laws_linvunitor_lunitor
    : ∏ (a b : strict_fiber_bicat_data)
         (f : strict_fiber_bicat_data ⟦ a, b ⟧),
       linvunitor f • lunitor f = id₂ f.
  Show proof.
    intros a b f ; cbn.
    rewrite disp_mor_transportf_prewhisker.
    rewrite !transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite !transport_f_f.
    etrans.
    {
      apply maponpaths.
      rewrite disp_vassocl.
      do 2 apply maponpaths.
      rewrite disp_vassocr.
      apply maponpaths, maponpaths_2.
      apply (disp_vcomp_linv (disp_local_iso_cleaving_invertible_2cell h (id_disp a;; f) (idempunitor c))).
    }
    unfold transportb.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      do 2 apply maponpaths.
      apply disp_id2_left.
    }
    unfold transportb.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      apply maponpaths.
      apply disp_linvunitor_lunitor.
    }
    unfold transportb.
    rewrite transport_f_f.
    apply (transportf_set (λ α : id₁ c ==> id₁ c, _ ==>[ α] _) _).
    apply C.

  Definition strict_fiber_bicat_data_laws_runitor_rinvunitor
    : ∏ (a b : strict_fiber_bicat_data)
         (f : strict_fiber_bicat_data ⟦ a, b ⟧),
       runitor f • rinvunitor f = id₂ (f · id₁ b).
  Show proof.
    intros a b f ; cbn.
    rewrite disp_mor_transportf_prewhisker.
    rewrite !transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite !transport_f_f.
    etrans.
    {
      apply maponpaths.
      rewrite disp_vassocl.
      do 2 apply maponpaths.
      rewrite disp_vassocr.
      apply maponpaths, maponpaths_2.
      apply disp_runitor_rinvunitor.
    }
    unfold transportb.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      do 2 apply maponpaths.
      apply disp_id2_left.
    }
    unfold transportb.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      apply maponpaths.
      apply (disp_vcomp_rinv
               (disp_local_iso_cleaving_invertible_2cell h (f ;; id_disp b)
                                                         (idempunitor c))).
    }
    unfold transportb.
    rewrite transport_f_f.
    apply (transportf_set (λ α : id₁ c ==> id₁ c, _ ==>[ α] _) _).
    apply C.

  Definition strict_fiber_bicat_data_laws_rinvunitor_runitor
    : ∏ (a b : strict_fiber_bicat_data)
         (f : strict_fiber_bicat_data ⟦ a, b ⟧),
       rinvunitor f • runitor f = id₂ f.
  Show proof.
    intros a b f ; cbn.
    rewrite disp_mor_transportf_prewhisker.
    rewrite !transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite !transport_f_f.
    etrans.
    {
      apply maponpaths.
      rewrite disp_vassocl.
      do 2 apply maponpaths.
      rewrite disp_vassocr.
      apply maponpaths, maponpaths_2.
      apply (disp_vcomp_linv (disp_local_iso_cleaving_invertible_2cell h (f ;; id_disp b) (idempunitor c))).
    }
    unfold transportb.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    rewrite disp_mor_transportf_postwhisker.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      do 2 apply maponpaths.
      apply disp_id2_left.
    }
    unfold transportb.
    rewrite disp_mor_transportf_prewhisker.
    rewrite transport_f_f.
    etrans.
    {
      apply maponpaths.
      apply disp_rinvunitor_runitor.
    }
    unfold transportb.
    rewrite transport_f_f.
    apply (transportf_set (λ α : id₁ c ==> id₁ c, _ ==>[ α] _) _).
    apply C.

  Definition strict_fiber_bicat_data_laws_lassociator_rassociator
    : ∏ (a₁ a₂ a₃ a₄ : strict_fiber_bicat_data)
         (f₁ : strict_fiber_bicat_data ⟦ a₁ , a₂ ⟧)
         (f₂ : strict_fiber_bicat_data ⟦ a₂ , a₃ ⟧)
         (f₃ : strict_fiber_bicat_data ⟦ a₃ , a₄ ⟧),
       lassociator f₁ f₂ f₃ • rassociator f₁ f₂ f₃
       =
       id₂ (f₁ · (f₂ · f₃)).
  Show proof.
    intros a₁ a₂ a₃ a₄ f₁ f₂ f₃ ; cbn.
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply disp_mor_transportf_prewhisker.
      }
      apply maponpaths.
      apply disp_mor_transportf_postwhisker.
    }
    do 2 refine (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)
                                _ _ _ _ _ _ @ _).
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocl _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths.
              refine (disp_vassocr _ _ _ @ _).
              etrans.
              {
                apply maponpaths.
                etrans.
                {
                  apply maponpaths_2.
                  refine (disp_vassocr _ _ _ @ _).
                  etrans.
                  {
                    apply maponpaths.
                    etrans.
                    {
                      apply maponpaths_2.
                      refine (disp_vassocr _ _ _ @ _).
                      etrans.
                      {
                        apply maponpaths.
                        etrans.
                        {
                          apply maponpaths_2.
                          exact (disp_vcomp_linv
                                   (disp_local_iso_cleaving_invertible_2cell
                                      h
                                      (local_iso_cleaving_1cell
                                         h (f₁;; f₂) (idempunitor c);; f₃)
                                      (idempunitor c))).
                        }
                        apply disp_mor_transportf_postwhisker.
                      }
                      etrans.
                      {
                        apply maponpaths.
                        etrans.
                        {
                          apply maponpaths.
                          apply disp_id2_left.
                        }
                        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                      }
                      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                    }
                    apply disp_mor_transportf_postwhisker.
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply disp_mor_transportf_postwhisker.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_prewhisker.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      refine (disp_vassocl _ _ _ @ _).
      etrans.
      {
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths.
            refine (disp_vassocr _ _ _ @ _).
            etrans.
            {
              apply maponpaths.
              etrans.
              {
                apply maponpaths_2.
                refine (disp_vassocr _ _ _ @ _).
                etrans.
                {
                  apply maponpaths.
                  etrans.
                  {
                    apply maponpaths_2.
                    refine (disp_rwhisker_vcomp _ _ @ _).
                    etrans.
                    {
                      apply maponpaths.
                      etrans.
                      {
                        apply maponpaths.
                        exact (disp_vcomp_linv
                                 (disp_local_iso_cleaving_invertible_2cell
                                    h (f₁;; f₂) (idempunitor c))).
                      }
                      apply disp_rwhisker_transport_left_new.
                    }
                    etrans.
                    {
                      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                    }
                    etrans.
                    {
                      apply maponpaths.
                      apply disp_id2_rwhisker.
                    }
                    apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                  }
                  etrans.
                  {
                    apply disp_mor_transportf_postwhisker.
                  }
                  etrans.
                  {
                    apply maponpaths.
                    apply disp_id2_left.
                  }
                  apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
                }
                apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
              }
              apply disp_mor_transportf_postwhisker.
            }
            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
          }
          apply disp_mor_transportf_prewhisker.
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        apply maponpaths.
        refine (disp_vassocr _ _ _ @ _).
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths_2.
            apply disp_lassociator_rassociator.
          }
          apply disp_mor_transportf_postwhisker.
        }
        etrans.
        {
          apply maponpaths.
          etrans.
          {
            apply maponpaths.
            apply disp_id2_left.
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
      }
      apply disp_mor_transportf_prewhisker.
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      etrans.
      {
        refine (disp_vassocl _ _ _ @ _).
        apply maponpaths.
        etrans.
        {
          apply maponpaths.
          refine (disp_vassocr _ _ _ @ _).
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              apply disp_lwhisker_vcomp.
            }
            apply disp_mor_transportf_postwhisker.
          }
          etrans.
          {
            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
          }
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths_2.
              etrans.
              {
                apply maponpaths.
                exact (disp_vcomp_rinv
                         (disp_local_iso_cleaving_invertible_2cell
                            h (f₂;; f₃) (idempunitor c))).
              }
              etrans.
              {
                apply disp_rwhisker_transport_right.
              }
              etrans.
              {
                apply maponpaths.
                apply disp_lwhisker_id2.
              }
              apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
            }
            apply disp_mor_transportf_postwhisker.
          }
          etrans.
          {
            apply maponpaths.
            etrans.
            {
              apply maponpaths.
              apply disp_id2_left.
            }
            apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
          }
          apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
        }
        apply disp_mor_transportf_prewhisker.
      }
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    etrans.
    {
      apply maponpaths.
      exact (disp_vcomp_rinv
               (disp_local_iso_cleaving_invertible_2cell
                  h
                  (f₁;; local_iso_cleaving_1cell h (f₂;; f₃) (idempunitor c))
                  (idempunitor c))).
    }
    etrans.
    {
      apply (@transport_f_f _ (λ z : _ ==> _, _ ==>[ z ] _)).
    }
    apply (transportf_set (λ α : id₁ c ==> id₁ c, _ ==>[ α] _) _).
    apply C.
End Strict_Fiber_Bicat.