Proposition spans_double_cat_hor_comp_laws : hor_comp_laws spans_double_cat_hor_comp_data. Proof. split. - intros a₁ a₂ a₃ h₁ h₂. use span_sqr_eq. use (MorphismsIntoPullbackEqual (isPullback_Pullback (PC _ _ _ _ _))) ; cbn. + unfold mor_of_comp_span_mor. rewrite PullbackArrow_PullbackPr1 ; cbn. rewrite id_left, id_right. apply idpath. + unfold mor_of_comp_span_mor. rewrite PullbackArrow_PullbackPr2 ; cbn. rewrite id_left, id_right. apply idpath. - intros. use span_sqr_eq. use (MorphismsIntoPullbackEqual (isPullback_Pullback (PC _ _ _ _ _))) ; cbn. + rewrite !assoc'. unfold mor_of_comp_span_mor. rewrite !PullbackArrow_PullbackPr1 ; cbn. rewrite !assoc. rewrite !PullbackArrow_PullbackPr1. rewrite !assoc'. apply idpath. + rewrite !assoc'. unfold mor_of_comp_span_mor. rewrite !PullbackArrow_PullbackPr2 ; cbn. rewrite !assoc. rewrite !PullbackArrow_PullbackPr2. rewrite !assoc'. 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.