Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms.
Referring to the above commutative diagram, one observes that every morphism
to emphasize its functorial nature, or sometimes merely in lower-case:
Categories that possess an internal Hom functor are referred to as closed categories. One has that