Homomorphism on Induced Structure to Commutative Semigroup

Theorem
Let $\struct {S, \circ}$ be an algebraic structure.

Let $\struct {T, \oplus}$ 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 pointwise operation on $T^S$ induced by $\oplus$.

Then $f \oplus' g$ is a homomorphism from $\struct {S, \circ}$ into $\struct {T, \oplus}$.

Proof
Let $\struct {T, \oplus}$ be a commutative semigroup.

Let $x, y \in S$.

Then:

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