Proposition fiberwise_univalent_2_0_disp_bicat_comp_cat : fiberwise_univalent_2_0 disp_bicat_comp_cat. Proof. refine (λ (C : cat_with_terminal_cleaving) (χ₁ χ₂ : comprehension_functor C), _). use weqhomot. - exact (disp_nat_z_iso_weq_disp_adjequiv χ₁ χ₂ ∘ disp_functor_eq_weq χ₁ χ₂ (disp_univalent_disp_codomain C) ∘ path_sigma_hprop _ _ _ (isaprop_is_cartesian_disp_functor _))%weq. - intro p ; cbn in p. induction p. use subtypePath. { intro. use (isaprop_disp_left_adjoint_equivalence _ _ is_univalent_2_1_bicat_cat_with_terminal_cleaving). exact disp_univalent_2_1_disp_bicat_comp_cat. } use disp_nat_trans_eq. intros x xx. use subtypePath. { intro. apply homset_property. } apply idpath. 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.