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Definition (Nonunital Strict Monoidal Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories (Definition A nonunital strict monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with the following properties:

  • The diagram of functors

    \[ \xymatrix { \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\ar [r]^-{\otimes } \ar [d]^{F \times F} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}\times \operatorname{\mathcal{D}}\ar [r]^-{ \otimes } & \operatorname{\mathcal{D}}} \]

    is strictly commutative. In particular, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have an equality $F(X) \otimes F(Y) = F(X \otimes Y)$ of objects of $\operatorname{\mathcal{D}}$.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the functor $F$ carries the associativity constraint $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$ (for the monoidal structure on $\operatorname{\mathcal{C}}$) to the associativity constraint $\alpha _{F(X), F(Y), F(Z) }: F(X) \otimes (F(Y) \otimes F(Z) ) \simeq (F(X) \otimes F(Y) ) \otimes F(Z)$ (for the monoidal structure on $\operatorname{\mathcal{D}}$).