Homomorphism on Induced Structure to Commutative Semigroup

Theorem
Let $$\left({S, \circ}\right)$$ be an algebraic structure.

Let $$\left({T, \oplus}\right)$$ be a commutative semigroup.

Let $$f$$ and $$g$$ be homomorphisms from $$S$$ into $$T$$.

Let $$f \oplus' g$$ be the operation on $T^S$ induced by $\oplus$.

Then $$f \oplus' g$$ is a homomorphism from $$\left({S, \circ}\right)$$ into $$\left({T, \oplus}\right)$$.

Proof
Let $$\left({T, \oplus}\right)$$ be a commutative semigroup.

Let $$x, y \in S$$. Then:

$$ $$ $$ $$

Notice that for this to work, $$\oplus$$ needs to be both commutative and associative.