Definition double_cat_are_conjoints {C : double_cat} {x y : C} (h : y -->h x) (v : x -->v y) : UU := ∑ (η : square v (identity_v x) (identity_h x) h) (ε : square (identity_v y) v h (identity_h y)), (transportf_square (id_v_left _) (id_v_left _) (rinvunitor_h h ⋆v transportf_square (id_v_right _) (id_v_right _) ((ε ⋆h η) ⋆v lunitor_h h)) = id_v_square h) × (transportf_square (idpath _) (id_v_left _) (transportf_square (id_v_right _) (idpath _) (η ⋆v ε)) = 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.