Definition sigma_disp_univalent_2_0_with_props (HD₁_2_0 : disp_univalent_2_0 D₁) (HD₂_2_0 : disp_univalent_2_0 D₂) : disp_univalent_2_0 (sigma_bicat _ _ D₂). Proof. apply fiberwise_univalent_2_0_to_disp_univalent_2_0. intros x xx yy. use weqhomot. - exact (sigma_idtoiso_2_0_alt HD₁_2_0 HD₂_2_0 xx yy). - intros p. cbn in p. induction p. use subtypePath. { intro ; apply isaprop_disp_left_adjoint_equivalence. + exact (pr2 HC). + apply sigma_disp_univalent_2_1_with_props ; assumption. } 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.PrecategoryBinProduct. 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.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.