Definition pair_disp_invertible_2cell_weq {a b : C} {f : a --> b} {g : a --> b} (x : invertible_2cell f g) {aa : disp_dirprod_bicat a} {bb : disp_dirprod_bicat b} (ff : aa -->[ f ] bb) (gg : aa -->[ g ] bb) : (disp_invertible_2cell x (pr1 ff) (pr1 gg) × disp_invertible_2cell x (pr2 ff) (pr2 gg)) ≃ disp_invertible_2cell x ff gg. Proof. use make_weq. - exact (pair_disp_invertible_2cell x). - use isweq_iso. + intros H. split. * apply pr1_disp_invertible_2cell. exact H. * apply pr2_disp_invertible_2cell. exact H. + intros H ; cbn. use total2_paths2. * use subtypePath. ** intro ; simpl. apply isaprop_is_disp_invertible_2cell. ** reflexivity. * use subtypePath. ** intro ; simpl. apply isaprop_is_disp_invertible_2cell. ** reflexivity. + intros H ; cbn. use subtypePath. * intro xx ; simpl. apply (@isaprop_is_disp_invertible_2cell C disp_dirprod_bicat). * 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.