Definition 4.43.2. Let $\mathcal{C}$ and $\mathcal{C}'$ be monoidal categories. A functor of monoidal categories $F : \mathcal{C} \to \mathcal{C}'$ is given by a functor $F$ as indicated and a isomorphism
\[ F(X) \otimes F(Y) \to F(X \otimes Y) \]
functorial in $X$ and $Y$ such that for all objects $X$, $Y$, and $Z$ the diagram
\[ \xymatrix{ F(X) \otimes (F(Y) \otimes F(Z)) \ar[r] \ar[d] & F(X) \otimes F(Y \otimes Z) \ar[r] & F(X \otimes (Y \otimes Z)) \ar[d] \\ (F(X) \otimes F(Y)) \otimes F(Z) \ar[r] & F(X \otimes Y) \otimes F(Z) \ar[r] & F((X \otimes Y) \otimes Z) } \]
commutes and such that $F(\mathbf{1})$ is a unit in $\mathcal{C}'$.
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