Definition pair_disp_adjequiv_to_sigma_disp_adjequiv {x : C} (xx : (sigma_bicat C D₁ D₂) x) (yy : (sigma_bicat C D₁ D₂) x) : (∑ (p : disp_adjoint_equivalence (internal_adjoint_equivalence_identity x) (pr1 xx) (pr1 yy)), disp_adjoint_equivalence (invmap (adjoint_equivalence_total_disp_weq (pr1 xx) (pr1 yy)) (internal_adjoint_equivalence_identity x ,, p)) (pr2 xx) (pr2 yy)) → disp_adjoint_equivalence (internal_adjoint_equivalence_identity x) xx yy. Proof. intros p. simple refine (_ ,, ((_ ,, (_ ,, _)) ,, _ ,, (_ ,, _))). - exact (pr11 p ,, pr12 p). - exact (pr112 (pr1 p) ,, pr112 (pr2 p)). - exact (pr1 (pr212 (pr1 p)) ,, pr1 (pr212 (pr2 p))). - exact (pr2 (pr212 (pr1 p)) ,, pr2 (pr212 (pr2 p))). - apply help_disp_left_adjoint_axioms. apply disp_2cells_isaprop_sigma ; assumption. - apply disp_locally_groupoid_sigma ; assumption. - apply disp_locally_groupoid_sigma ; assumption. 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.