Homomorphism to Group Preserves Identity

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

Let $\left({T, *}\right)$ be a group.

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

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

Then:
 * $\phi \left({e_S}\right) = e_T$

Proof
By hypothesis, $\left({T, *}\right)$ is a group.

By the Cancellation Laws, all elements of $T$ are cancellable.

Thus Homomorphism with Cancellable Codomain Preserves Identity can be applied.