Isomorphism Preserves Identity

Theorem
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism.

Then $\circ$ has an identity $e_S$ $\map \phi {e_S}$ is the identity for $*$.