Pointwise Operation is Composite of Operation with Mapping to Cartesian Product

Theorem
Let $S$ be a set.

Let $\struct {T, *}$ be an algebraic structure.

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

Let the algebraic structure $\struct {T^S, \oplus}$ be the algebraic structure on $T^S$ induced by $*$.

Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings.

Let $f \times g : S \to T \times T$ be the mapping from $S$ to the cartesian product $T \times T$ defined by:
 * $\forall x \in S : \map {\paren {f \times g}} x = \tuple {\map f x, \map g x}$

Then:
 * $f \oplus g = * \circ \paren {f \times g}$

That is, $f \oplus g$ is the composition of the binary operation $*$ with the mapping $f \times g: S \to T \times T$.

Proof
From Equality of Mappings:
 * $* \circ \paren {f \times g} = f \oplus g$