The latter equation is expressed by the following commutative diagram: In this situation, one can show that formula_13 can be turned into a functor formula_36 in a unique way such that  formula_37  for all morphisms formula_38 in formula_22; formula_15 is then called a left adjoint to formula_13. Similarly, we may define right-adjoint functors. A functor formula_42 is a right adjoint functor if for each object formula_30 in formula_28,  there exists a universal morphism from formula_30 to formula_13. Spelled out, this means that for each object formula_30 in formula_28,  there exists an object formula_49 in formula_22 and a morphism formula_51 such that for every object formula_21 in formula_22  and every morphism formula_33 there exists a unique morphism formula_32 with formula_56. Again, this formula_15 can be uniquely turned into a functor formula_20 such that formula_59 for formula_60 a morphism in formula_28; formula_13 is then called a right adjoint to formula_15. It is true, as the terminology implies, that formula_15 is left adjoint to formula_13 if and only if formula_13 is right adjoint to formula_15. These definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem. A hom-set adjunction between two categories "C" and "D" consists of two functors "F" : "D" → "C" and and a natural isomorphism This specifies a family of bijections for all objects "X" in "C" and "Y" in "D". In this situation, "F" is left adjoint to "G" and "G" is right adjoint to "F" . This definition is a logical compromise in that it is somewhat more difficult to satisfy than the universal morphism definitions, and has fewer immediate implications than the counit–unit definition. It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions. In order to interpret Φ as a "natural isomorphism", one must recognize and as functors. In fact, they are both bifunctors from to Set (the category of sets).