Proposition structured_cospans_double_cat_hor_comp_laws : hor_comp_laws structured_cospans_double_cat_hor_comp_data. Proof. split. - intros a₁ a₂ a₃ h₁ h₂. use struct_cospan_sqr_eq. use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX _ _ _ _ _))) ; cbn. + unfold mor_of_comp_struct_cospan_mor. rewrite PushoutArrow_PushoutIn1 ; cbn. rewrite id_left, id_right. apply idpath. + unfold mor_of_comp_struct_cospan_mor. rewrite PushoutArrow_PushoutIn2 ; cbn. rewrite id_left, id_right. apply idpath. - intros. use struct_cospan_sqr_eq. use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX _ _ _ _ _))) ; cbn. + rewrite !assoc. unfold mor_of_comp_struct_cospan_mor. rewrite !PushoutArrow_PushoutIn1 ; cbn. rewrite !assoc'. rewrite !PushoutArrow_PushoutIn1. rewrite !assoc. apply idpath. + rewrite !assoc. unfold mor_of_comp_struct_cospan_mor. rewrite !PushoutArrow_PushoutIn2 ; cbn. rewrite !assoc'. rewrite !PushoutArrow_PushoutIn2. rewrite !assoc. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. 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.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.PseudoDoubleSetCats. Require Import UniMath.Bicategories.DoubleCategories.Core.CompanionsAndConjoints.