Proposition ver_opposite_double_cat_pentagon : pentagon_law ver_opposite_double_cat_associator. Proof. intros v w x y z h₁ h₂ h₃ h₄ ; cbn. pose (rassociator_rassociator (C := horizontal_bicat C) h₁ h₂ h₃ h₄) as p. cbn in p. refine (_ @ maponpaths (transportb_square (id_v_left _ @ id_v_left _) (id_v_left _ @ id_v_left _)) (!p) @ _). - unfold transportb_square. rewrite transportf_f_square. use transportf_square_eq. apply idpath. - unfold transportb_square. rewrite transportf_square_prewhisker. rewrite !transportf_f_square. etrans. { apply maponpaths. refine (!_). apply square_assoc_v'. } 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.