# 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