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.

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    $\begingroup$ You could also get this transformation with a braided monoidal category (but I don't know if this would make $\otimes $ strong monoidal). $\endgroup$ – Arnaud D. Feb 20 at 20:01
  • $\begingroup$ @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. $\endgroup$ – John Gowers Feb 20 at 20:39

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