Remark 4.3.2.7. Let $\operatorname{Cat}$ denote the category of (small) categories. Then $\operatorname{Cat}$ admits a monoidal structure, where the tensor product is given by the join functor
\[ \star : \operatorname{Cat}\times \operatorname{Cat}\rightarrow \operatorname{Cat}\quad \quad (\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \mapsto \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}} \]
of Remark 4.3.2.3, and the associativity constraints are the isomorphisms of Remark 4.3.2.6. The unit for this monoidal structure is the empty category $\emptyset \in \operatorname{Cat}$ (Example 4.3.2.4).