Definition iso_twosided_disp_to_z_iso_disp {C₁ C₂ : category} {D : twosided_disp_cat C₁ C₂} {x : C₁} {y : C₂} {xx yy : D x y} (ff : iso_twosided_disp (identity_z_iso x) (identity_z_iso y) xx yy) : @z_iso_disp _ (twosided_disp_cat_to_disp_cat _ _ D) (x ,, y) (x ,, y) (identity_z_iso _) xx yy. Proof. simple refine (_ ,, _ ,, _ ,, _). - exact (pr1 ff). - exact (iso_inv_twosided_disp (pr2 ff)). - abstract (refine (iso_after_inv_twosided_disp (pr2 ff) @ _) ; cbn ; refine (!_) ; apply transportb_dirprodeq). - abstract (refine (inv_after_iso_twosided_disp (pr2 ff) @ _) ; cbn ; refine (!_) ; apply transportb_dirprodeq). 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.Core.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat.