Definition 4.42.11. 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

$F(X) \otimes F(Y) \to F(X \otimes Y)$

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

$\xymatrix{ F(X) \otimes F(Y) \ar[r] \ar[d] & F(X \otimes Y) \ar[d] \\ F(Y) \otimes F(X) \ar[r] & F(Y \otimes X) }$

commutes.

There are also:

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