Proposition psfunctor_psfunctor_bicat_to_fib_comp_axioms {F G H : psfunctor (cat_to_bicat C^op) bicat_of_univ_cats} (n₁ : pstrans F G) (n₂ : pstrans G H) : disp_nat_trans_axioms (psfunctor_psfunctor_bicat_to_fib_comp_data n₁ n₂). Proof. intros x y f xx yy ff ; cbn -[psfunctor_id psfunctor_comp]. refine (!_). etrans. { apply maponpaths. apply maponpaths_2. do 3 apply maponpaths. refine (id_left _ @ _). apply maponpaths. refine (id_left _ @ _). apply id_right. } etrans. { apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat H)). } etrans. { do 2 apply maponpaths_2. refine (!_). apply (nat_trans_ax (pr1 (psfunctor_id H x))). } refine (!_). etrans. { do 3 apply maponpaths_2. apply functor_comp. } do 3 refine (assoc' _ _ _ @ _). refine (!_). do 3 refine (assoc' _ _ _ @ _). apply maponpaths. refine (assoc' _ _ _ @ _). do 2 apply maponpaths. etrans. { refine (assoc _ _ _ @ _). apply maponpaths_2. refine (_ @ !(nat_trans_eq_pointwise (psfunctor_rinvunitor H f) _)). refine (_ @ assoc _ _ _). exact (!(id_left _)). } etrans. { apply maponpaths_2. apply psfunctor_idtoiso. } refine (!(pr1_idtoiso_concat _ _) @ _). refine (!_). etrans. { refine (_ @ !(nat_trans_eq_pointwise (psfunctor_linvunitor H f) _)). refine (_ @ assoc _ _ _). exact (!(id_left _)). } etrans. { apply psfunctor_idtoiso. } do 2 apply maponpaths. refine (_ @ maponpathscomp0 (λ z, pr1 (#H z) _) _ _). apply maponpaths. apply homset_property. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedCategory. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedFunctor. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedTransformation. Require Import UniMath.CategoryTheory.IndexedCategories.FibrationToIndexedCategory. Require Import UniMath.CategoryTheory.IndexedCategories.CartesianToIndexedFunctor. Require Import UniMath.CategoryTheory.IndexedCategories.NatTransToIndexed. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedCategoryToFibration. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedFunctorToCartesian. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedTransformationToTransformation. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Discreteness. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.DiscreteBicat. Require Import UniMath.Bicategories.Core.Examples.FibSlice. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.PseudoFunctorsIntoCat. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Transformations.Examples.PseudoTransformationIntoCat. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.Modifications.Examples.ModificationIntoCat.