Proposition spans_double_cat_triangle : triangle_law spans_double_cat_lunitor spans_double_cat_runitor spans_double_cat_associator. Proof. intro ; intros. use span_sqr_eq. rewrite transportb_disp_mor2_span ; cbn. use (MorphismsIntoPullbackEqual (isPullback_Pullback (PC _ _ _ _ _))) ; cbn. - unfold span_associator_mor, mor_of_comp_span_mor ; cbn. rewrite !assoc'. rewrite !PullbackArrow_PullbackPr1. rewrite !assoc. rewrite PullbackArrow_PullbackPr1. unfold span_runitor_mor. rewrite PullbackArrow_PullbackPr1, id_right. apply idpath. - unfold span_associator_mor, mor_of_comp_span_mor ; cbn. rewrite !assoc'. rewrite !PullbackArrow_PullbackPr2. rewrite !assoc. rewrite PullbackArrow_PullbackPr2. refine (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.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Spans. Require Import UniMath.CategoryTheory.Limits.Pullbacks. 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.