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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