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Remark (Composition of Monoidal Functors). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors equipped with nonunital lax monoidal structures $\mu $ and $\nu $, respectively, so that the composite functor $G \circ F$ inherits a nonunital lax monoidal structure (Construction If $\mu $ and $\nu $ are monoidal structures on $F$ and $G$, then $G \circ F$ inherits a monoidal structure. This observation (and its counterpart for monoidal natural transformations) imply that the composition law of Construction restricts to a functor

\[ \circ : \operatorname{Fun}^{\otimes }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}). \]