Definition are_companions {B : verity_double_bicat} {x y : B} (h : x --> y) (v : x -|-> y) : UU := ∑ (φ : square_double_bicat h (id_h y) v (id_v y)) (ψ : square_double_bicat (id_h x) h (id_v x) v), (runitor _ ▹s (linvunitor _ ◃s ψ ⋆v φ) = id_h_square_bicat _) × (runitor _ ▿s (linvunitor _ ▵s ψ ⋆h φ) = id_v_square_bicat _).
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.