Library UniMath.Bicategories.PseudoFunctors.Preservation.PullbackPreservation

Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.Bicategories.Core.Bicat.
Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.Morphisms.Adjunctions.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.Slice.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Import PseudoFunctor.Notations.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.PullbackFunctor.
Require Import UniMath.Bicategories.PseudoFunctors.Biadjunction.
Require Import UniMath.Bicategories.PseudoFunctors.Preservation.Preservation.
Require Import UniMath.Bicategories.Modifications.Modification.
Require Import UniMath.Bicategories.Colimits.Initial.
Require Import UniMath.Bicategories.Colimits.Extensive.
Require Import UniMath.Bicategories.Colimits.Examples.SliceBicategoryColimits.
Require Import UniMath.Bicategories.Limits.Pullbacks.
Require Import UniMath.Bicategories.Limits.PullbackFunctions.

Local Open Scope cat.

1. Pullbacks preserve strict biinitial objects

Definition pullback_preserves_biinitial
           (B : bicat_with_pb)
           (HI : strict_biinitial_obj B)
           {b₁ b₂ : B}
           (f : b₁ --> b₂)
  : preserves_biinitial
      (pb_psfunctor B f).
Show proof.

  pose (H := map_to_strict_biinitial_is_biinitial
               (pr2 HI)
               (pb_pr2 f (is_biinitial_1cell_property (pr12 HI) b₂))).
  use (preserves_chosen_biinitial_to_preserve_biinitial
         (_ ,, _)
         (pb_psfunctor B f)).
  - apply biinitial_in_slice.
    exact (pr1 HI ,, pr12 HI).
  - use (equiv_from_biinitial
           (is_biinitial_slice
              (pb_obj f (is_biinitial_1cell_property (pr12 HI) b₂)
               ,,
               H)
              b₁)).
    + use make_1cell_slice.
      * apply id₁.
      * cbn.
        use is_biinitial_invertible_2cell_property.
        exact H.
    + use left_adjoint_equivalence_in_slice_bicat.
      cbn.
      apply internal_adjoint_equivalence_identity.