# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 2.1.4.18. In the situation of Construction 2.1.4.17, suppose that $\mu$ and $\nu$ are nonunital monoidal structures on $F$ and $G$, respectively: that is, assume that all of the tensor constraints

$\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F( X \otimes Y) \quad \quad \nu _{U,V}: G(U) \otimes G(V) \rightarrow G( U \otimes V)$

are isomorphisms. Then Construction 2.1.4.17 supplies a nonunital monoidal structure on the composite functor $G \circ F$. We therefore obtain a composition law

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