Proposition psfunctor_fib_to_psfunctor_counit_data_ob_data_functor_laws (F : psfunctor (cat_to_bicat C^op) bicat_of_univ_cats) (x : C) : is_functor (psfunctor_fib_to_psfunctor_counit_data_ob_data_functor_data F x). Proof. split. - intros xx ; cbn -[psfunctor_id]. exact (nat_trans_eq_pointwise (vcomp_rinv (psfunctor_id F x)) xx). - intros xx yy zz ff gg ; cbn -[psfunctor_id psfunctor_comp]. cbn in xx, yy, zz, ff, gg. rewrite !assoc. apply maponpaths_2. refine (!_). rewrite !assoc'. etrans. { apply maponpaths. exact (!(nat_trans_ax ((psfunctor_id F x)^-1) _ _ gg)). } assert (# F (identity x) ◃ (psfunctor_id F x)^-1 = psfunctor_comp F (identity x) (identity x) • ##F (runitor _) • rinvunitor _) as p. { rewrite (psfunctor_F_runitor F (identity x)). rewrite !vassocr. rewrite vcomp_rinv. rewrite id2_left. rewrite !vassocl. rewrite runitor_rinvunitor. rewrite id2_right. apply idpath. } pose (q := nat_trans_eq_pointwise p zz). cbn -[psfunctor_id psfunctor_comp] in q. rewrite id_right in q. etrans. { do 2 apply maponpaths. exact q. } refine (!_). etrans. { apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat F)). } rewrite !assoc'. do 3 apply maponpaths. refine (_ @ !(psfunctor_idtoiso F _ _)). do 3 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.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. 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. Require Import UniMath.Bicategories.Grothendieck.FibrationToPseudoFunctor. Require Import UniMath.Bicategories.Grothendieck.PseudoFunctorToFibration.