As I understand things, the desire to find a more categorical construction of the free abelian group on a set S is not itself fully in the categorical spirit. Rather, for an object which is defined by a universal mapping property, you only need to convince yourself that it exists, not to be bothered by looking at any particular construction. The point is that anything that you care about this object will follow most transparently from the universal mapping property.
In this case, if I have a universal map S -> FreeAb(S) and a map of sets S -> T, we can define a map FreeAb(S) -> FreeAb(T) without "looking under the hood" at how FreeAb's are constructed. Namely, by composing S -> T and T -> FreeAb(T) we get a set map S -> FreeAb(T). By the universal mapping property, this factors through a homomorphism FreeAb(S) -> FreeAb(T), which is what we wanted. Done!