Proposition psfunctor_psfunctor_bicat_to_fib_id_laws (F : psfunctor (cat_to_bicat C^op) bicat_of_univ_cats) : disp_nat_trans_axioms (psfunctor_psfunctor_bicat_to_fib_id_data F). Proof. intros x y f xx yy ff ; cbn -[psfunctor_id psfunctor_comp]. pose (p := nat_trans_eq_pointwise (psfunctor_linvunitor F f) yy). cbn -[psfunctor_id psfunctor_comp] in p. rewrite id_left in p. rewrite assoc'. etrans. { apply maponpaths. exact (!p). } refine (!_). etrans. { apply maponpaths. apply maponpaths_2. etrans. { do 2 apply maponpaths. etrans. { apply maponpaths. apply id_right. } apply id_right. } exact (!(nat_trans_ax (pr1 (psfunctor_id F x)) _ _ ff)). } cbn -[psfunctor_id psfunctor_comp]. pose (q := nat_trans_eq_pointwise (psfunctor_rinvunitor F f) yy). cbn -[psfunctor_id psfunctor_comp] in q. rewrite assoc'. etrans. { do 2 apply maponpaths. refine (_ @ !q). apply maponpaths_2. exact (!(id_left _)). } etrans. { apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat F)). } rewrite !assoc'. apply maponpaths. etrans. { apply maponpaths_2. apply psfunctor_idtoiso. } refine (_ @ !(psfunctor_idtoiso F _ _)). refine (!(pr1_idtoiso_concat _ _) @ _). do 2 apply maponpaths. refine (!(maponpathscomp0 (λ h, pr1 (#F h) yy) _ _) @ _). 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.