Variant 1.3.2.8. A *nonunital monoid* is a set $M$ equipped with a map

\[ m: M \times M \rightarrow M \quad \quad (x,y) \mapsto xy \]

which satisfies the associative law $x(yz) = (xy)z$ for $x,y,z \in M$. If $M$ and $M'$ are nonunital monoids, a function $f: M \rightarrow M'$ is a *nonunital monoid homomorphism* if it satisfies the equation $f(xy) = f(x) f(y)$ for every pair of elements $x,y \in M$. We let $\operatorname{Mon}^{\operatorname{nu}}$ denote the category whose objects are nonunital monoids and whose morphisms are nonunital monoid homomorphisms.