Definition sigma_is_univalent_2_1 (HC_2_1 : is_univalent_2_1 C) (HD₁_2_1 : disp_univalent_2_1 D₁) (HD₂_2_1 : disp_univalent_2_1 D₂) : is_univalent_2_1 E₂. Proof. intros x y f g. use weqhomot. - exact (idtoiso_2_1_alt_E₂ f g HC_2_1 HD₁_2_1 HD₂_2_1). - intros p. induction p ; cbn. use subtypePath. { intro. apply (@isaprop_is_invertible_2cell (total_bicat (sigma_bicat C D₁ D₂))). } 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.