Proposition structured_cospans_double_cat_lunitor_laws : double_lunitor_laws structured_cospans_double_cat_lunitor_data. Proof. intros x₁ x₂ y₁ y₂ h₁ h₂ v₁ v₂ sq. use struct_cospan_sqr_eq. rewrite transportb_disp_mor2_struct_cospan ; cbn. use (MorphismsOutofPushoutEqual (isPushout_Pushout (PX _ _ _ _ _))) ; cbn. - rewrite !assoc. unfold struct_cospan_lunitor_mor, mor_of_comp_struct_cospan_mor. rewrite !PushoutArrow_PushoutIn1. rewrite !assoc'. rewrite PushoutArrow_PushoutIn1. exact (struct_cospan_sqr_mor_left _ sq). - rewrite !assoc. unfold struct_cospan_lunitor_mor, mor_of_comp_struct_cospan_mor. rewrite !PushoutArrow_PushoutIn2. rewrite !assoc'. rewrite PushoutArrow_PushoutIn2. rewrite id_left, id_right. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. 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.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.PseudoDoubleSetCats. Require Import UniMath.Bicategories.DoubleCategories.Core.CompanionsAndConjoints.