Homomorphism with Cancellable Codomain Preserves Identity

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

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

If every element of $\left({T, *}\right)$ is cancellable, then $\phi \left({e_S}\right)$ is the identity for $*$.

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

Let $\left({T, *}\right)$ be such that every element is cancellable.

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

As every element of $\left({T, *}\right)$ is cancellable, then from Identity Only Idempotent Cancellable Element, if there is an idempotent element in $\left({T, *}\right)$, it must be an identity.

Thus:

So $\phi \left({e_S}\right)$ is idempotent in $\left({T, *}\right)$ and the result follows.

Also see

 * Homomorphism with Identity Preserves Inverses