Proposition structured_cospans_double_cat_functor_comp_iso_inv_laws : is_inverse_in_precat (struct_cospan_sqr_ob_mor L₂ (lax_double_functor_comp_h structured_cospans_double_cat_functor h k)) structured_cospans_double_cat_functor_comp_iso_inv. Proof. split ; unfold structured_cospans_double_cat_functor_comp_iso_inv. - use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX₂ _ _ _ _ _))) ; cbn. + rewrite !assoc. rewrite PushoutArrow_PushoutIn1. rewrite id_right. apply (PushoutArrow_PushoutIn1 P). + rewrite !assoc. rewrite PushoutArrow_PushoutIn2. rewrite id_right. apply (PushoutArrow_PushoutIn2 P). - use (MorphismsOutofPushoutEqual (isPushout_Pushout P)) ; cbn. + rewrite !assoc. rewrite (PushoutArrow_PushoutIn1 P). rewrite PushoutArrow_PushoutIn1. rewrite id_right. apply idpath. + rewrite !assoc. rewrite (PushoutArrow_PushoutIn2 P). rewrite PushoutArrow_PushoutIn2. rewrite id_right. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. 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.Examples.StructuredCospans. Require Import UniMath.CategoryTheory.Limits.Pushouts. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Examples.StructuredCospansDoubleCat.