Proposition ver_opposite_double_cat_triangle : triangle_law ver_opposite_double_cat_lunitor ver_opposite_double_cat_runitor ver_opposite_double_cat_associator. Proof. intros x y z h k ; cbn. pose (triangle_r_inv (C := horizontal_bicat C) k h) as p. cbn in p. rewrite !comp_h_square_id in p. rewrite square_id_left_v in p. rewrite square_id_right_v in p. unfold transportb_square in p. rewrite transportf_f_square in p. refine (!_). refine (_ @ maponpaths (transportb_square (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. 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.