abstract algebra – Mapping a product into a local ring


All rings considered are unitial commutative. I have been bugged by the following claim.

Let $R=A times B$ be product of two rings. Suppose $S$ is a local ring.
Then the canonical map induced from projections
$$ Hom(A,S) sqcup Hom(B,S) rightarrow Hom(R,S) $$
is surjective.


I understand:

But can’t really see how I can put the secondition into use. Any hint would be appreciated