Proposition are_companions_right' {B : verity_double_bicat} {x y : B} {h : x --> y} {v : x -|-> y} (c : are_companions h v) : counit_are_companions c ⋆h unit_are_companions c = rinvunitor _ ▿s (lunitor _ ▵s id_v_square_bicat _). Proof. rewrite <- (are_companions_right c). rewrite !dwhisker_uwhisker_square. rewrite <- uwhisker_square_comp. rewrite lunitor_linvunitor. rewrite uwhisker_square_id. rewrite <- dwhisker_square_comp. rewrite runitor_rinvunitor. rewrite dwhisker_square_id. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.DerivedLaws. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CellsAndSquares. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.UnderlyingCats. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.LocalUnivalence. Require Import UniMath.Bicategories.DoubleCategories.Examples.TransposeDoubleBicat.