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Definition Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. A monoidal structure on $F$ is a nonunital lax monoidal structure $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ (Definition which satisfies the following additional conditions:

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the tensor constraint $\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F(X \otimes Y)$ is an isomorphism in $\operatorname{\mathcal{D}}$ (that is, $\mu $ is a nonunital monoidal structure on $F$).

  • There exists an isomorphism $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \xrightarrow {\sim } F(\mathbf{1}_{\operatorname{\mathcal{C}}})$ which is a unit for $F$ (in the sense of Definition

A monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a pair $(F, \mu )$, where $F$ is a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ and $\mu $ is a monoidal structure on $F$.