Definition structured_cospans_double_cat_functor_comp_data : double_functor_hor_comp_data (twosided_disp_cat_of_struct_cospans_functor α) (structured_cospans_double_cat_hor_comp PX₁ L₁) (structured_cospans_double_cat_hor_comp PX₂ L₂). Proof. intros x y z h k. use make_struct_cospan_sqr. - use PushoutArrow ; cbn. + exact (#FX (PushoutIn1 _)). + exact (#FX (PushoutIn2 _)). + abstract (rewrite !assoc' ; apply maponpaths ; rewrite <- !functor_comp ; apply maponpaths ; apply PushoutSqrCommutes). - abstract (split ; [ cbn ; rewrite !assoc' ; rewrite functor_id, id_left ; rewrite PushoutArrow_PushoutIn1 ; rewrite <- !functor_comp ; apply idpath | cbn ; rewrite !assoc' ; rewrite functor_id, id_left ; rewrite PushoutArrow_PushoutIn2 ; rewrite <- !functor_comp ; apply idpath ]). Defined.
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.