Proposition is_z_iso_struct_cospan_runitor_mor_eqs : is_inverse_in_precat struct_cospan_runitor_mor (PushoutIn1 (comp_struct_cospan_Pushout h (id_struct_cospan y))). Proof. split. - unfold struct_cospan_runitor_mor. use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX _ _ _ _ _))) ; cbn. + rewrite !assoc. rewrite PushoutArrow_PushoutIn1. rewrite id_left, id_right. apply idpath. + rewrite !assoc. rewrite PushoutArrow_PushoutIn2. rewrite PushoutSqrCommutes. rewrite id_left, id_right. apply idpath. - unfold struct_cospan_runitor_mor. rewrite PushoutArrow_PushoutIn1. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.Limits.Pushouts. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Total. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Constant. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.DispCatOnTwoSidedDispCat.