proposition: Proposition spans_double_cat_triangle : triangle_law spans_double_cat_lunitor spans_double_cat_runitor spans_double_cat_associator.
proof: 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.
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.Strictness UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Spans UniMath.CategoryTheory.Limits.Pullbacks 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