Definition 4.43.11. Let $\mathcal{C}$ and $\mathcal{C}'$ be symmetric monoidal categories. A *functor of symmetric monoidal categories* $F : \mathcal{C} \to \mathcal{C}'$ is given by a functor $F$ as indicated and an isomorphism

functorial in $X$ and $Y$ such that $F$ is a functor of monoidal categories and such that for all objects $X$ and $Y$ the diagram

commutes.

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