Definition disp_presheaves_adjequiv_weq {C : ∁} (FC FC' : disp_presheaf_bicat C) : @invertible_2cell bicat_of_univ_cats _ _ FC FC' ≃ disp_adjoint_equivalence (internal_adjoint_equivalence_identity C) FC FC'. Proof. exists (disp_presheaves_adjequiv FC FC'). use isweq_iso. - exact (disp_presheaves_adjequiv_inv FC FC'). - intro x. apply subtypePath. { intro ; apply isaprop_is_invertible_2cell. } reflexivity. - intro x. apply subtypePath. { intro. apply isaprop_disp_left_adjoint_equivalence. + apply univalent_cat_is_univalent_2_1. + apply disp_presheaves_is_univalent_2_1. } reflexivity. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.