Lemma bidisp_unit_disp_prebicat_is_globally_univalent : disp_univalent_2_0 bidisp_unit_disp_bicat. Proof. intros C D equalcats IC ID. induction equalcats. use weqhomot. - set (i1 := iso_dispadjequiv_equivalence C IC ID). set (i3 := (_ ,, (pr2 C) IC ID)). exact (i1 ∘ i3)%weq. - intro p. induction p; cbn. use subtypePath. + intro; simpl. apply (@isaprop_disp_left_adjoint_equivalence bicat_of_univ_cats bidisp_unit_disp_bicat). * exact univalent_cat_is_univalent_2_1. * exact 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.