Remark 4.3.3.4 (Functoriality). Construction 4.3.3.2 determines a functor
\[ \star : \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \times \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \quad \quad (X,Y) \mapsto X \star Y. \]
Note that this functor preserves colimits separately in each variable.