Remark 2.1.6.13 (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 2.1.4.17). 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 2.1.4.17 restricts to a functor
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
\[ \circ : \operatorname{Fun}^{\otimes }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}). \]