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. Then:

Let $$x, y \in S$$. Then:

$$ $$ $$ $$ $$