# For symmetric monoidal categories, the monoidal product is a monoidal functor

Has anyone got a reference for the following fact?

If $$\mathcal X$$ is a symmetric monoidal category, then $$\_\otimes\_\colon\mathcal X\times\mathcal X \to \mathcal X$$ is a strong monoidal functor. Moreover, the associators and unitors in $$\mathcal X$$ are monoidal natural transformations.

The key point here is the existence of a natural transformation $$(a \otimes b) \otimes (c \otimes d) \to (a \otimes c) \otimes (b \otimes d)\,,$$ which is why we need symmetry.

• You could also get this transformation with a braided monoidal category (but I don't know if this would make $\otimes$ strong monoidal). – Arnaud D. Feb 20 at 20:01
• @ArnaudD. My guess is that this is still true for braided monoidal categories, though it's harder to show since we can't immediately apply the coherence theorem. – John Gowers Feb 20 at 20:39