Kerodon

Example 2.1.4.12. Let $\operatorname{Set}$ denote the category of sets, endowed with the monoidal structure given by cartesian product of sets (Example 2.1.3.2). For each set $S$, we can identify nonunital algebra structures on $S$ (in the sense of Example 2.1.4.11) with nonunital monoid structures on $S$ (in the sense of Variant 1.3.2.8). This observation supplies an isomorphism of categories $\operatorname{Alg}^{\operatorname{nu}}( \operatorname{Set}) \simeq \operatorname{Mon}^{\operatorname{nu}}$, where $\operatorname{Mon}^{\operatorname{nu}}$ is the category of nounital monoids.