Proposition hor_opposite_double_cat_pentagon : pentagon_law hor_opposite_double_cat_associator. Proof. intros v w x y z h₁ h₂ h₃ h₄ ; cbn. pose (rassociator_rassociator (C := horizontal_bicat C) h₄ h₃ h₂ h₁) as p. cbn in p. simple refine (_ @ maponpaths (transportb_square (id_v_right _ @ id_v_right _) (id_v_right _ @ id_v_right _)) (!p) @ _). - unfold transportb_square. rewrite transportf_f_square. rewrite transportb_disp_mor2_hor_op. use transportf_disp_mor2_eq. apply idpath. - unfold transportb_square. rewrite transportf_square_prewhisker. rewrite !transportf_f_square. apply transportf_square_id. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.opp_precat. 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.Opposites. 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.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. 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.DerivedLaws.TransportLaws. Require Import UniMath.Bicategories.DoubleCategories.DerivedLaws.Variations. Require Import UniMath.Bicategories.DoubleCategories.Underlying.HorizontalBicategory.