Definition pair_disp_adjequiv_to_sigma_disp_adjequiv_inv_pr1 {x : C} (xx : (sigma_bicat C D₁ D₂) x) (yy : (sigma_bicat C D₁ D₂) x) : disp_adjoint_equivalence (internal_adjoint_equivalence_identity x) xx yy → disp_adjoint_equivalence (internal_adjoint_equivalence_identity x) (pr1 xx) (pr1 yy). Proof. intro p. simple refine (pr11 p ,, ((_ ,, (_ ,, _)) ,, _ ,, (_ ,, _))). - exact (pr1 (pr112 p)). - exact (pr11 (pr212 p)). - exact (pr12 (pr212 p)). - apply help_disp_left_adjoint_axioms. exact HD₁. - apply LG₁. - apply LG₁. 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.