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

functorial in $X$ and $Y$ such that for all objects $X$, $Y$, and $Z$ the diagram

commutes and such that $F(\mathbf{1})$ is a unit in $\mathcal{C}'$.

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