Functor between Monoid Categories

Theorem
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be monoids.

Let $\mathbf S$ and $\mathbf T$ be the associated monoid categories.

Let $F: \mathbf S \to \mathbf T$ be a functor.

Then the object functor $F_0$ of $F$ is a monoid homomorphism.