Remark 2.1.5.20 (00DV): Closure under Composition

Remark 2.1.5.20 (Closure under Composition). 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 $F$ and $G$ admit units

\[ \delta : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \quad \quad \epsilon : \mathbf{1}_{\operatorname{\mathcal{E}}} \rightarrow G( \mathbf{1}_{\operatorname{\mathcal{D}}} ), \]

then the composite map

\[ \mathbf{1}_{\operatorname{\mathcal{E}}} \xrightarrow { \epsilon } G( \mathbf{1}_{\operatorname{\mathcal{D}}} ) \xrightarrow { G( \delta )} (G \circ F)( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \]

is a unit for the composite functor $G \circ F$. This observation (and its counterpart for monoidal natural transformations) imply that the composition law of Construction 2.1.4.17 restricts to a functor

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