Proposition are_conjoints_left' {B : verity_double_bicat} {x y : B} {h : x --> y} {v : y -|-> x} (c : are_conjoints h v) : counit_are_conjoints c ⋆v unit_are_conjoints c = linvunitor _ ▹s (runitor _ ◃s id_h_square_bicat _). Proof. rewrite <- (are_conjoints_left c). rewrite !rwhisker_lwhisker_square. rewrite <- lwhisker_square_comp. rewrite runitor_rinvunitor. rewrite lwhisker_square_id. rewrite <- rwhisker_square_comp. rewrite lunitor_linvunitor. rewrite rwhisker_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.DoubleBicat.CompanionPairs. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CompanionPairUnique. Require Import UniMath.Bicategories.DoubleCategories.Examples.HorizontalDual.