Induced Structure Identity

Theorem
Let $$\left({T, \oplus}\right)$$ be an algebraic structure, and let $$S$$ be a set.

Let $$\left({T^S, \oplus}\right)$$ be the structure on $$T^S$$ induced by $$\oplus$$.

If $$e$$ is an identity for $$\oplus$$, then the mapping $$f_e: S \to T$$ defined as:

$$\forall x \in S: f_e \left({x}\right) = e$$

is the identity for the operation induced on $$T^S$$ by $$\oplus$$.

Proof
Let $$\left({T, \oplus}\right)$$ be a structure with an identity $$e$$.

Let $$f \in T^S$$. Then:

$$ $$ $$ $$ $$ $$