Lemma iso_dispadjequiv_equivalence (C : bicat_of_univ_cats) (IC ID : bidisp_unit_disp_bicat C) : z_iso IC ID ≃ disp_adjoint_equivalence (idtoiso_2_0 C C (idpath C)) IC ID. Proof. use make_weq. - apply iso_to_dispadjequiv. - use isweq_iso. + apply dispadjequiv_to_iso. + intro i. induction i. use subtypePath. * intro f. apply isaprop_is_z_isomorphism. * apply idpath. + intro adjequiv. induction adjequiv. use subtypePath. * intro f. apply isaprop_disp_left_adjoint_equivalence. -- apply univalent_cat_is_univalent_2_1. -- apply bidisp_unit_disp_prebicat_is_locally_univalent. * apply idpath. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories.