Let A, B and C be three objects in the category Set. For simplicity, assume that their underlying sets contain a finite number of elements, a, b and c respectively. Using the usual Haskell notation for types, there exists exactly (c^a)^(c^b×b^a) morphisms from the object/type (b->c, a->b) to the object/type a->c. Among this potentially huge set of morphisms, exactly one is the (uncurried) composition function, defined as compose (g, f) = g•f.

My question is, does this particular morphism have any universal property that make it possible to identify using only category theory?