Inverse Mapping in Induced Structure of Homomorphism to Abelian Group

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

Let $\left({T, \oplus}\right)$ be an abelian group.

Let $f$ be a homomorphism from $S$ into $T$.

Let $f^*$ be the induced structure inverse of $f$.

Then $f^*$ is a homomorphism from $\left({S, \circ}\right)$ into $\left({T, \oplus}\right)$.

Proof
Let $\left({T, \oplus}\right)$ be an abelian group.

Let $x, y \in S$.

Then: