Proposition structured_cospans_double_cat_functor_comp_laws : double_functor_hor_comp_laws structured_cospans_double_cat_functor_comp_data. Proof. intro ; intros. use struct_cospan_sqr_eq. rewrite transportb_disp_mor2_struct_cospan ; cbn. use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX₂ _ _ _ _ _))) ; cbn. - unfold mor_of_comp_struct_cospan_mor. rewrite !assoc. rewrite !PushoutArrow_PushoutIn1. cbn. rewrite <- !functor_comp. rewrite !PushoutArrow_PushoutIn1. rewrite !assoc'. rewrite !PushoutArrow_PushoutIn1. rewrite <- !functor_comp. apply idpath. - unfold mor_of_comp_struct_cospan_mor. rewrite !assoc. rewrite !PushoutArrow_PushoutIn2. cbn. rewrite <- !functor_comp. rewrite !PushoutArrow_PushoutIn2. rewrite !assoc'. rewrite !PushoutArrow_PushoutIn2. rewrite <- !functor_comp. 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.