Lemma iso_to_dispadjequiv (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. intro i. induction i as [finv [f [li ri]]]. split with f. unfold disp_left_adjoint_equivalence. repeat (use tpair); try apply univalent_category_has_homsets. - exact finv. - etrans. { apply id_left. } exact (! li). - etrans. { apply cancel_postcomposition. apply ri. } apply id_right. - etrans. { apply cancel_postcomposition. apply li. } apply id_left. - etrans. { apply id_left. } exact (! ri). 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.