Identity of Monoid is Unique

Theorem
Let $\left({S, \circ}\right)$ be a semigroup that has an identity $e \in S$.

Then $e$ is unique.

Proof
As $\left({S, \circ}\right)$ is an algebraic structure, the result Identity is Unique can be applied directly.

Also see

 * A semigroup with an identity is called a monoid.