Definition disp_adjequiv_sigma_help_weq (x : C) (xx1 : D₁ x) (xx2 yy2 : D₂ (x,, xx1)) : disp_adjoint_equivalence (@internal_adjoint_equivalence_identity (total_bicat D₁) (x,, xx1)) xx2 yy2 ≃ disp_adjoint_equivalence (invmap (adjoint_equivalence_total_disp_weq xx1 xx1) ((internal_adjoint_equivalence_identity x) ,,disp_identity_adjoint_equivalence xx1)) xx2 yy2. Proof. use make_weq. - exact (disp_adjequiv_sigma_help x xx1 xx2 yy2). - use isweq_iso. + exact (disp_adjequiv_sigma_help_inv x xx1 xx2 yy2). + intros p. use subtypePath. { intro ; apply isaprop_disp_left_adjoint_equivalence. + apply total_is_univalent_2_1. * exact (pr2 HC). * exact HD₁_2_1. + exact HD₂_2_1. } apply idpath. + intros p. use subtypePath. { intro ; apply isaprop_disp_left_adjoint_equivalence. + apply total_is_univalent_2_1. * exact (pr2 HC). * exact HD₁_2_1. + exact HD₂_2_1. } 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.