Definition pair_adjoint_equivalence_weq {a b : C} (HC : is_univalent_2_1 C) (HD1 : disp_univalent_2_1 D1) (HD2 : disp_univalent_2_1 D2) (f : adjoint_equivalence a b) (aa : disp_dirprod_bicat a) (bb : disp_dirprod_bicat b) : (disp_adjoint_equivalence f (pr1 aa) (pr1 bb) × disp_adjoint_equivalence f (pr2 aa) (pr2 bb)) ≃ (disp_adjoint_equivalence f aa bb). Proof. use tpair. - exact (pair_adjoint_equivalence f aa bb). - use isweq_iso. + intros H. split. * exact (pr1_adjoint_equivalence f aa bb H). * exact (pr2_adjoint_equivalence f aa bb H). + intros A. use total2_paths2. * use subtypePath. ** intro ; simpl. apply isaprop_disp_left_adjoint_equivalence. *** exact HC. *** exact HD1. ** reflexivity. * use subtypePath. ** intro ; simpl. apply isaprop_disp_left_adjoint_equivalence. *** exact HC. *** exact HD2. ** reflexivity. + intros H ; cbn. use subtypePath. * intro xx ; simpl. apply (@isaprop_disp_left_adjoint_equivalence C disp_dirprod_bicat). ** exact HC. ** exact (is_univalent_2_1_dirprod_bicat HD1 HD2). * reflexivity. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. 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.Core.Examples.Initial. Require Import UniMath.Bicategories.Core.Examples.Final. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.