Epimorphism Preserves Identity

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

If $$\circ$$ has an identity $$e_S$$, 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 $$e_S$$.

Then $$\forall x \in S: x \circ e_S = x = e_S \circ x$$.

The result follows directly from the morphism property of $$\circ$$ under $$\phi$$:

$$ $$

Comment
Note that this result is applied to epimorphisms. For a general homomorphism which is not surjective, we can say nothing definite about the behaviour of the elements of its codomain which are not part of its image.

However, also see: for when the algebraic structure is actually a group.
 * Group Homomorphism Preserves Identity