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Remark In the situation of Construction, 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 supplies a nonunital monoidal structure on the composite functor $G \circ F$. We therefore obtain a composition law

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