Proposition cell_are_companions_id {B : verity_double_bicat} (H : vertically_saturated B) {x y : B} {h : x --> y} {v : x -|-> y} (c : are_companions h v) : cell_are_companions H (id2 h) c c = id2 v. Proof. unfold cell_are_companions. rewrite <- (square_to_vertical_cell_id H). apply maponpaths. rewrite dwhisker_square_id. rewrite are_companions_left'. rewrite <- rwhisker_square_comp. rewrite rinvunitor_runitor. rewrite rwhisker_square_id. rewrite <- lwhisker_square_comp. rewrite linvunitor_lunitor. rewrite lwhisker_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.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. 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.CompanionPairs.