Homomorphism with Identity Preserves Inverses

Theorem
Let $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ be a homomorphism.

Let $$\left({S, \circ}\right)$$ have an identity $$e_S$$.

Let $$\left({T, *}\right)$$ also have an identity $$e_T = \phi \left({e_S}\right)$$.

If $$x^{-1}$$ is an inverse of $$x$$ for $$\circ$$, then $$\phi \left({x^{-1}}\right)$$ is an inverse of $$\phi \left({x}\right)$$ for $$*$$. That is, $$\phi \left({x^{-1}}\right) = \left({\phi \left({x}\right)}\right)^{-1}$$.

Proof
Let $$\left({S, \circ}\right)$$ be an algebraic structure in which $$\circ$$ has an identity $$e_S$$.

Let $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ be a homomorphism.

Let $$\left({T, *}\right)$$ be an algebraic structure in which $$*$$ has an identity $$e_T = \phi \left({e_S}\right)$$.

Using the morphism property:

$$ $$ $$ $$