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 ($b\rightarrow c$, $a\rightarrow b$) to the object/type $a\rightarrow c$. Among this potentially huge set of morphisms, exactly one is the (uncurried) composition function, defined as ${\rm compose}\, (g, f) = g\circ f$.

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

This is the computer-scientist-toying-with-categories-version of the question. If you prefer a version written by a proper mathematician, please refer to The universal property of composition of morphisms