Let $S$ and $T$ be two arbitrary sets and consider the free vector space $C(S)$ and $C(T)$ generated respectively by $S$ and $T$. Show that $C(S \times T)$ is isomorphic to $C(S) \otimes C(T)$.
I know that in this link: $C(S \times T)$ is isomorphic to $C(S) \otimes C(T).$ there is an answer to this question. But I want to solve the problem in a different way.
My approach is: I want to construct a bilinear map $\varphi:C(S) \times C(T) \rightarrow C(S) \otimes C(T)$ and then by the universal property exists a linear map $ f: C(S) \otimes C(T) \rightarrow C(S \times T) $ such that $\varphi = f \circ \otimes$ and i want $f$ to be bijection. So, i think that $\varphi(s,t) = (s,t)$ and $f(s \otimes t) = (s,t) $ but i think that these maps does not work.
Any Suggestion? Thanks for your help.