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.