proposition: Proposition is_z_iso_struct_cospan_lunitor_mor_eqs : is_inverse_in_precat struct_cospan_lunitor_mor (PushoutIn2 (comp_struct_cospan_Pushout (id_struct_cospan x) h)).
proof: Proof. split. - unfold struct_cospan_lunitor_mor. use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX _ _ _ _ _))) ; cbn. + rewrite !assoc. rewrite PushoutArrow_PushoutIn1. rewrite <- PushoutSqrCommutes. rewrite id_left, id_right. apply idpath. + rewrite !assoc. rewrite PushoutArrow_PushoutIn2. rewrite id_left, id_right. apply idpath. - unfold struct_cospan_lunitor_mor. rewrite PushoutArrow_PushoutIn2. apply idpath. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Total UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Constant UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.DispCatOnTwoSidedDispCat