Definition 1.3.2.3. Let $M$ and $M'$ be monoids having unit elements $e$ and $e'$, respectively. A function $f: M \rightarrow M'$ is a monoid homomorphism if it satisfies the identities
\[ f(e) = e' \quad \quad f( xy ) = f(x) f(y) \]
for every pair of elements $x,y \in M$. We let $\operatorname{Mon}$ denote the category whose objects are monoids and whose morphisms are monoid homomorphisms.