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.
See Proposition 2 (page 13) of Joyal–Street's Braided monoidal categories. (Perhaps also relevant is Roberts's A braided monoidal category is symmetric if and only if the product is braided.)