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 $T^S$ be the set of all mappings from $S$ to $T$.

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 associative and commutative.