proposition: Proposition structured_cospans_double_cat_functor_runitor : double_functor_runitor (structured_cospans_double_cat_runitor PX₁ L₁) (structured_cospans_double_cat_runitor PX₂ L₂) structured_cospans_double_cat_functor_id structured_cospans_double_cat_functor_comp.
proof: Proof. intros x y f. use struct_cospan_sqr_eq. rewrite transportf_disp_mor2_struct_cospan ; cbn. unfold struct_cospan_runitor_mor, mor_of_comp_struct_cospan_mor. use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX₂ _ _ _ _ _))) ; cbn. - rewrite !assoc. rewrite !PushoutArrow_PushoutIn1. rewrite !assoc'. rewrite (maponpaths (λ z, _ · z) (assoc _ _ _)). rewrite PushoutArrow_PushoutIn1. rewrite <- functor_comp. rewrite PushoutArrow_PushoutIn1. rewrite functor_id, id_left. apply idpath. - rewrite !assoc. rewrite !PushoutArrow_PushoutIn2. rewrite !assoc'. rewrite (maponpaths (λ z, _ · z) (assoc _ _ _)). rewrite PushoutArrow_PushoutIn2. rewrite <- functor_comp. rewrite PushoutArrow_PushoutIn2. 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