Closed category

In a locally small, the external hom (x, y) maps a pair of objects to a set of morphisms.

In the same vein, in a closed category, the (object of) morphisms from one object to another can be seen as lying inside the category.

Every closed category has a forgetful functor to the category of sets, which in particular takes the internal hom to the external hom.

with a so-called internal Hom functor with left Yoneda arrows natural in

with a natural isomorphism and a dinatural transformation all satisfying certain coherence conditions.