Definition adjequiv_in_E₂_weq (x y : E₂) (HC_2_1 : is_univalent_2_1 C) (HD₁_2_1 : disp_univalent_2_1 D₁) (HD₂_2_1 : disp_univalent_2_1 D₂) : adjoint_equivalence (E₂_to_E₁ x) (E₂_to_E₁ y) ≃ adjoint_equivalence x y. Proof. use make_weq. - exact (adjequiv_in_E₂ x y). - use isweq_iso. + exact (adjequiv_in_E₂_inv x y). + intros l. use subtypePath. { intro. apply isaprop_left_adjoint_equivalence. apply E₁_univalent_2_1; assumption. } apply idpath. + intros l. use subtypePath. { intro. apply isaprop_left_adjoint_equivalence. apply sigma_is_univalent_2_1; assumption. } 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.