Quotient Structure of Monoid is Monoid

Theorem
Let $\mathcal R$ be a congruence relation on a monoid $\left({S, \circ}\right)$ with an identity $e$.

Then the quotient structure $\left({S / \mathcal R, \circ_\mathcal R}\right)$ is a monoid.

Proof
From Quotient Structure of Semigroup is Semigroup $\left({S / \mathcal R, \circ_\mathcal R}\right)$ is a semigroup.

Let $\left[\!\left[{x}\right]\!\right]_\mathcal R \in S / \mathcal R$.

Consider $\left[\!\left[{e}\right]\!\right]_\mathcal R$:

Furthermore:

Hence $\left[\!\left[{e}\right]\!\right]_\mathcal R$ is an identity.

Hence $\left({S / \mathcal R, \circ_\mathcal R}\right)$ is a monoid.