Unique Constant in Category of Monoids

Theorem
Let $\mathbf{Mon}$ be the category of monoids.

Then every object $M$ of $\mathbf{Mon}$ has precisely one constant.

Corollary
The category of monoids $\mathbf{Mon}$ does not have enough constants.

Proof
From Trivial Monoid is Terminal Element, we obtain that a constant of $M$ is a morphism $f: \left\{{e}\right\} \to M$.

By Trivial Monoid is Initial Element, there is precisely one such morphism.

Hence $M$ has one constant.