Definition pair_disp_invertible_2cell {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. intros H. use tpair. - split. + apply H. + apply H. - apply pair_is_disp_invertible_2cell. split. + exact (pr2 (pr1 H)). + exact (pr2 (pr2 H)). 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.