Proposition disp_univalent_2_1_disp_bicat_comp_cat : disp_univalent_2_1 disp_bicat_comp_cat. Proof. use fiberwise_local_univalent_is_univalent_2_1. refine (λ (C₁ C₂ : cat_with_terminal_cleaving) (F : functor_with_terminal_cleaving C₁ C₂) (χ₁ : comprehension_functor C₁) (χ₂ : comprehension_functor C₂) (Fχ₁ Fχ₂ : comprehension_nat_trans χ₁ χ₂ F), _). use isweqimplimpl. - intros p ; cbn. use disp_nat_trans_eq. intros x xx. use subtypePath. { intro. apply homset_property. } refine (_ @ pr1 p x xx @ _) ; cbn. + refine (!_). refine (_ @ id_right _). apply maponpaths. apply comprehension_functor_mor_id. + apply id_left. - cbn -[isaprop]. unfold comprehension_nat_trans in *. use (isaset_disp_nat_trans _ (disp_functor_composite χ₁ (disp_codomain_functor F)) (disp_functor_composite (comp_cat_type_functor F) χ₂)). - use isaproptotal2. + intro. apply isaprop_is_disp_invertible_2cell. + intros. apply isaprop_comprehension_nat_trans_eq. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.ComprehensionC. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.FibTerminal. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat.