Definition disp_locally_groupoid_sigma (LG₁ : disp_locally_groupoid D₁) (LG₂ : disp_locally_groupoid D₂) : disp_locally_groupoid (sigma_bicat C D₁ D₂). Proof. use make_disp_locally_groupoid_univalent_2_1. - intros a b f aa bb ff gg xx. pose (p₁ := pr1 xx ,, LG₁ _ _ _ _ _ _ _ _ _ (pr1 xx) : disp_invertible_2cell _ _ _). pose (pr2 xx) as m. cbn in m. pose (p₂ := pr2 xx ,, LG₂ (a ,, pr1 aa) (b ,, pr1 bb) (f ,, pr1 ff) (f ,, pr1 gg) (iso_in_E_weq _ _ (id2_invertible_2cell f ,, p₁)) (pr2 aa) (pr2 bb) (pr2 ff) (pr2 gg) m). exact (pr2 (pair_disp_invertible_to_sigma_disp_invertible _ _ (p₁ ,, p₂))). - exact (pr2 HC). 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.