Remark 4.3.3.6. The join operation of Construction 4.3.3.2 determines a functor
\[ \star : \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \times \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}). \]
This functor determines a monoidal structure on the category $\operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$, whose associativity constraints are the isomorphisms $\alpha _{X,Y,Z}$ of Remark 4.3.3.5 and whose unit object is the functor $E$ of Example 4.3.3.3.