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This is probably a most stupid question, but I really do not have a profound knowledge of monoidal and monoidal braided categories, I only skimmed across them in a course on Hopf Algebras.

My question is: can we always give a monoidal category a (somehow canonical) braiding, as uninteresting and useless it might be? If not, can we charachterize those monoidal categories for which it is possible to do so? Are there, on the other hand (interesting) examples of monoidal categories which cannot be bradided?
All I found was more on examples of braided monoidal categories which are not symmetric, but that is a different matter.

Thanks in advance.

Francesco Bilotta
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    Take the monoidal category whose objects are the elements of a monoid, morphisms are just identities, and monoidal product is the product in the monoid. Then if the monoid doesn't commute there's no braiding. – Oscar Cunningham Jun 26 '20 at 10:58

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