Proposition psfunctor_fib_to_psfunctor_counit_data_ob_data_functor_inv_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_inv_data F x). Proof. split. - intros xx. cbn -[psfunctor_id]. apply id_left. - intros xx yy zz ff gg. cbn -[psfunctor_id psfunctor_comp]. refine (!_). etrans. { rewrite !assoc'. do 2 apply maponpaths. rewrite !assoc. apply maponpaths_2. refine (!_). apply (nat_trans_ax (pr1 (psfunctor_id F x))). } cbn -[psfunctor_id psfunctor_comp]. pose (nat_trans_eq_pointwise (psfunctor_rinvunitor F (identity x)) zz) as p. cbn -[psfunctor_id psfunctor_comp] in p. etrans. { do 2 apply maponpaths. rewrite !assoc'. do 2 apply maponpaths. refine (!_ @ !p). rewrite !assoc'. apply id_left. } etrans. { apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat F)). } rewrite !assoc'. do 2 apply maponpaths. refine (_ @ id_right _). apply maponpaths. etrans. { apply maponpaths_2. apply (psfunctor_idtoiso F _ _). } refine (!(pr1_idtoiso_concat _ _) @ _). refine (_ @ idtoiso_idpath _). do 2 apply maponpaths. refine (!(maponpathscomp0 (λ h, pr1 (#F h) zz) _ _) @ _ @ maponpaths_idpath). 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.