Lemma dispadjequiv_to_iso (C : bicat_of_univ_cats) (IC ID : bidisp_unit_disp_bicat C) : disp_adjoint_equivalence (idtoiso_2_0 C C (idpath C)) IC ID -> z_iso IC ID. Proof. intro equalunits. induction equalunits as [f adj]. induction adj as [data [ladj_ax weq_ax]]. induction data as [finv [dunit dcounit]]. cbn in *. unfold functor_identity in dunit. unfold nat_trans_id in dunit. cbn in dunit. exists finv. exists f. split. - etrans. { apply (pathsinv0 dunit). } apply id_left. - apply pathsinv0. etrans. { apply (pathsinv0 dcounit). } apply id_right. 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.