Definition pair_disp_adjequiv_to_sigma_disp_adjequiv_weq {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. use make_weq. - exact (pair_disp_adjequiv_to_sigma_disp_adjequiv xx yy). - use isweq_iso. + exact (pair_disp_adjequiv_to_sigma_disp_adjequiv_inv xx yy). + exact (pair_disp_adjequiv_to_sigma_disp_adjequiv_weq_help xx yy). + intros p. use subtypePath. { intro ; apply isaprop_disp_left_adjoint_equivalence ; [ exact (pr2 HC) | exact sigma_disp_univalent_2_1_with_props ]. } 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.