# Kerodon

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Construction 2.1.4.17 (Composition of Nonunital Monoidal Functors). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be nonunital monoidal categories, and suppose we are given a pair of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. If $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ is a nonunital lax monoidal structure on the functor $F$ and $\nu = \{ \nu _{U,V} \} _{U, V \in \operatorname{\mathcal{D}}}$ is a nonunital lax monoidal structure on $G$, then the composite functor $G \circ F$ inherits a nonunital lax monoidal structure, which associates to each pair of objects $X,Y \in \operatorname{\mathcal{C}}$ the composite map

$(G \circ F)(X) \otimes (G \circ F)(Y) \xrightarrow { \nu _{ F(X), F(Y)} } G( F(X) \otimes F(Y) ) \xrightarrow { G( \mu _{X,Y} ) } (G \circ F)(X \otimes Y).$

This construction determines a composition law

$\circ : \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}).$