Library UniMath.CategoryTheory.IndexedCategories.IndexedTransformationToTransformation

1. The data
2. The proof of naturality
  Proposition indexed_nat_trans_to_disp_nat_trans_axioms
    : disp_nat_trans_axioms indexed_nat_trans_to_disp_nat_trans_data.
  Show proof.
    intros x y f xx yy ff ; cbn in *.
    unfold transportb, indexed_nat_trans_to_disp_nat_trans_data.
    rewrite !functor_comp.
    refine (_ @ !(transportf_indexed_cat_to_disp_cat _ _ _)).
    rewrite !assoc'.
    etrans.
    {
      apply maponpaths.
      rewrite !assoc.
      do 2 apply maponpaths_2.
      exact (!(indexed_nat_trans_natural_inv m f yy)).
    }
    rewrite !assoc.
    etrans.
    {
      do 3 apply maponpaths_2.
      apply (nat_trans_ax (m x)).
    }
    rewrite !assoc'.
    apply maponpaths.
    refine (!_).
    etrans.
    {
      rewrite !assoc.
      do 2 apply maponpaths_2.
      etrans.
      {
        apply maponpaths_2.
        refine (!_).
        apply (nat_trans_ax (indexed_cat_id Ψ x)).
      }
      cbn.
      rewrite !assoc'.
      apply maponpaths.
      refine (!_).
      apply (nat_trans_ax (indexed_cat_id Ψ x)).
    }
    cbn.
    rewrite !assoc'.
    do 2 apply maponpaths.
    refine (_ @ !(indexed_cat_lunitor_alt Ψ _ _)).
    rewrite !assoc.
    etrans.
    {
      apply maponpaths_2.
      apply (indexed_cat_runitor_alt Ψ).
    }
    refine (!(pr1_idtoiso_concat _ _) @ _).
    do 2 apply maponpaths.
    refine (!(maponpathscomp0 (λ g, (Ψ $ g) (θ y yy)) _ _) @ _).
    apply maponpaths.
    apply homset_property.

3. The displayed natural transformation