proposition: Proposition structured_cospans_double_cat_lunitor_laws : double_lunitor_laws structured_cospans_double_cat_lunitor_data.
proof: 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.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories 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.DoubleCats UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats UniMath.Bicategories.DoubleCategories.Core.PseudoDoubleSetCats UniMath.Bicategories.DoubleCategories.Core.CompanionsAndConjoints