Kerodon

Remark 4.3.3.10. For every pair of functors $X,Y: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$, we have a canonical bijection $(X \star Y)(\emptyset ) = X(\emptyset ) \times Y(\emptyset )$. In particular, if $X$ and $Y$ belong to the subcategory $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \subseteq \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$, then the join $X \star Y$ also belongs to $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$. Moreover, $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ contains the unit object $E$ of Example 4.3.3.3. It follows that $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ inherits the structure of a monoidal category (with respect to the join operation of Construction 4.3.3.2).