proposition: Proposition structured_cospans_double_cat_functor_comp_laws : double_functor_hor_comp_laws structured_cospans_double_cat_functor_comp_data.
proof: 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.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.StructuredCospans UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.Preservation UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor UniMath.Bicategories.DoubleCategories.Core.DoubleCats UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats UniMath.Bicategories.DoubleCategories.Examples.StructuredCospansDoubleCat