Show that if the arrow $f: A \to B$ is an epimorphism then the arrow $id \times f: C\times A \to C \times B$ is an epimorphism as well

Question

I'm following the book Sets for Mathematics by Lawvere. As an exercise in section 7 it's stated that if we have an arrow $f: A \rightarrow B$ that is an epimorphism then the arrow $id\times f: C\times A \to C \times B$ is also an epimorphism. On the book it's recomended to do so by the use of exponentials. Maybe we can use the property that since $f$ is an epi then, for all $V$ it happens that the arrow $V^f: V^B \to V^A$ is injective. In particular the arrow $C^f: C^B \to C^A$, although I don't see how we could use it if in fact we could use it.