Definition disp_nat_z_iso_weq_disp_adjequiv : disp_nat_z_iso χ₁ χ₂ (nat_z_iso_id (functor_identity C)) ≃ disp_adjoint_equivalence (D := disp_bicat_comp_cat) (internal_adjoint_equivalence_identity C) χ₁ χ₂. Proof. use weq_iso. - exact disp_nat_z_iso_to_adjequiv. - exact adjequiv_to_disp_nat_z_iso. - abstract (intros τ ; use subtypePath ; [ intro ; apply isaprop_is_disp_nat_z_iso | ] ; use disp_nat_trans_eq ; intros x xx ; apply idpath). - abstract (intros τ ; 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 ; apply idpath). Defined.
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.