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In the course of checking something else, I needed to know that $\otimes \colon C\times C\to C$ is a braided monoidal functor when $(C,\otimes)$ is symmetric monoidal. I couldn’t find this, so proved it myself, then only now discovered this is essentially contained in Proposition 5.4 of “Braided tensor categories” (Joyal and Street, but the published version). However, at John B’s urging I had proved these two things are equivalent (categorifying the result that a group is abelian iff its multiplication is a homomorphism). The proof is very short (less than a page), much shorter than all other treatments I’ve seen (eg via $E_\infty$-algebras in the category of $E_\infty$-algebras in $\mathbf{Cat}$). For your amusement, it’s here. When I wrote this I had only consulted “Braided monoidal categories” (Joyal and Street, the preprint version), which I had always understood to contain strictly more than “Braided tensor categories”, hence the last paragraph.