Suppose $e$ and $e^{\prime }$ are identity elements. Then $e=e^{\prime }$.

Proof

$e\cdot e^{\prime }=e^{\prime }$, since $e$ is an identity
$e\cdot e^{\prime }=e$, since $e^{\prime }$ is an identity
so $e=e\cdot e^{\prime }=e^{\prime }$.

QED