proposition: Proposition is_iso_struct_cospan_associator_mor_eq : is_inverse_in_precat struct_cospan_associator_mor struct_cospan_associator_mor_inv.
proof: Proof. split. - use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX _ _ _ _ _))) ; cbn. + rewrite id_right. unfold struct_cospan_associator_mor. rewrite !assoc. rewrite PushoutArrow_PushoutIn1. unfold struct_cospan_associator_mor_inv. rewrite !assoc'. rewrite !PushoutArrow_PushoutIn1. apply idpath. + rewrite id_right. unfold struct_cospan_associator_mor. rewrite !assoc. rewrite PushoutArrow_PushoutIn2. use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX _ _ _ _ _))) ; cbn. * rewrite !assoc. rewrite PushoutArrow_PushoutIn1. unfold struct_cospan_associator_mor_inv. rewrite !assoc'. rewrite !PushoutArrow_PushoutIn1. rewrite !PushoutArrow_PushoutIn2. apply idpath. * rewrite !assoc. rewrite PushoutArrow_PushoutIn2. unfold struct_cospan_associator_mor_inv. rewrite !PushoutArrow_PushoutIn2. apply idpath. - use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX _ _ _ _ _))) ; cbn. + rewrite id_right. unfold struct_cospan_associator_mor_inv. rewrite !assoc. rewrite PushoutArrow_PushoutIn1. use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX _ _ _ _ _))) ; cbn. * rewrite !assoc. rewrite PushoutArrow_PushoutIn1. unfold struct_cospan_associator_mor. rewrite !PushoutArrow_PushoutIn1. apply idpath. * rewrite !assoc. rewrite PushoutArrow_PushoutIn2. unfold struct_cospan_associator_mor. rewrite !assoc'. rewrite PushoutArrow_PushoutIn2. rewrite PushoutArrow_PushoutIn1. apply idpath. + rewrite id_right. unfold struct_cospan_associator_mor_inv. rewrite !assoc. rewrite PushoutArrow_PushoutIn2. unfold struct_cospan_associator_mor. rewrite !assoc'. 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