> One little nuance: we're assuming we have _a_ coproduct for every pair of objects, and _an_ initial object. But we need to pick a specific choice of these to make \\(C\\) into a symmetric monoidal category, because the definition of symmetric monoidal category talks about _the_ tensor product of objects, and _the_ unit for the tensor product.
> Luckily, one can prove that different choices will make \\(C\\) into symmetric monoidal categories that are 'equivalent', using the precise definition of equivalence for symmetric monoidal categories (which is [Definition 13 here](http://math.ucr.edu/home/baez/qg-winter2001/definitions.pdf).)
That's a good point, I was treating the coproducts and initial objects as unique since even though they need not be unique, they must be isomorphic to one another. But for that reason, the different symmetric categories must be equivalent.