Isomorphism Preserves Identity

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

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

Then $\circ$ has an identity $e_S$ iff $\phi \left({e_S}\right)$ is the identity for $*$.