Definition double_cat_are_companions {C : double_cat} {x y : C} (h : x -->h y) (v : x -->v y) : UU := ∑ (φ : square v (identity_v y) h (identity_h y)) (ψ : square (identity_v x) v (identity_h x) h), (transportf_square (id_v_right _) (id_v_right _) (transportf_square (id_v_left _) (id_v_left _) (linvunitor_h h ⋆v (ψ ⋆h φ)) ⋆v runitor_h h) = id_v_square h) × (transportf_square (idpath _) (id_right _) (transportf_square (id_left _) (idpath _) (ψ ⋆v φ)) = id_h_square v).
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.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.DerivedLaws.TransportLaws. Require Import UniMath.Bicategories.DoubleCategories.Underlying.HorizontalBicategory.