Quotient Structure of Monoid is Monoid

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

Then the quotient structure $\struct {S / \mathcal R, \circ_\mathcal R}$ is a monoid.

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

Let $\eqclass x {\mathcal R} \in S / \mathcal R$.

Consider $\eqclass e {\mathcal R}$:

Furthermore:

Hence $\eqclass e {\mathcal R}$ is an identity.

Hence $\struct {S / \mathcal R, \circ_\mathcal R}$ is a monoid.