Let S be the set of generators of a free group and let G be an arbitrary group. Map S into G in any way you like, and the free group must follow. If S contains A B and C, then the word ABC in the free group has to map to h(A)h(B)h(C) in G. Concatenate words together in the free group, and join their images in G. Cancellation kills h(X)h(x) in G, which doesn't change anything. The map is a valid homomorphism, and it is the only possible group homomorphism consistent with h(S).
If S and T have the same cardinality, the free group on S is isomorphic to the free group on T. This is an immediate consequence of category theory, but if you like you can prove it by building a bijection between S and T, and the words of the two groups will correspond.