Definition disp_cell_unit_bicat_univalent_2_0 (HC : is_univalent_2_1 C) (HDP : ∏ (a b : C) (f : a --> b) (aa : D a) (bb : D b), isaprop (aa -->[ f] bb)) (Dset : ∏ (a : C), isaset (D a)) (inv : ∏ (a : C) (aa bb : disp_cell_unit_bicat a), (aa -->[ id₁ a ] bb × bb -->[ id₁ a ] aa) → aa = bb) : disp_univalent_2_0 disp_cell_unit_bicat. Proof. apply fiberwise_univalent_2_0_to_disp_univalent_2_0. intros a aa bb. use isweqimplimpl. - intro η ; cbn. apply inv. exact (invmap (disp_cell_unit_bicat_adjoint_equivalent (idtoiso_2_0 a a (idpath a)) aa bb) η). - apply (Dset a). - apply (isofhlevelweqf 1 (disp_cell_unit_bicat_adjoint_equivalent _ _ _)). apply isapropdirprod ; apply HDP. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.Initial. Require Import UniMath.Bicategories.Core.Examples.Final. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.