Definition 2.1.0.5. Let $M$ and $M'$ be nonunital monoids. We say that a function $f: M \rightarrow M'$ is a *nonunital monoid homomorphism* if, for every pair of elements $x,y \in M$, we have $f(xy) = f(x) f(y)$. If $M$ and $M'$ are monoids, we say that $f$ is a *monoid homomorphism* if it is a nonunital monoid homomorphism which carries the unit element $e \in M$ to the unit element $e' \in M'$.

We let $\operatorname{Mon}^{\operatorname{nu}}$ denote the category whose objects are nonunital monoids and whose morphisms are nonunital monoid homomorphisms, and $\operatorname{Mon}\subset \operatorname{Mon}^{\operatorname{nu}}$ the subcategory whose objects are monoids and whose morphisms are monoid homomorphisms.