Proposition structured_cospans_double_cat_functor_lunitor : double_functor_lunitor (structured_cospans_double_cat_lunitor PX₁ L₁) (structured_cospans_double_cat_lunitor PX₂ L₂) structured_cospans_double_cat_functor_id structured_cospans_double_cat_functor_comp. Proof. intros x y f. use struct_cospan_sqr_eq. rewrite transportf_disp_mor2_struct_cospan ; cbn. unfold struct_cospan_lunitor_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. apply idpath. - rewrite !assoc. rewrite !PushoutArrow_PushoutIn2. rewrite !assoc'. rewrite (maponpaths (λ z, _ · z) (assoc _ _ _)). rewrite PushoutArrow_PushoutIn2. rewrite <- functor_comp. rewrite PushoutArrow_PushoutIn2. rewrite functor_id. rewrite id_left. 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.