Scott wrote:

> **Theorem:** A category \$$C\$$ with all binary coproducts (there exists a coproduct for every pair of objects) and an initial object \$$\mathbf{0}\$$ is a monoidal category where the coproduct is the tensor product and the initial object is the unit. If for any pair of objects \$$x,y\$$, there exists a pair of morphisms \$$f : x \to y\$$ and \$$f' : y \to x\$$, then \$$C\$$ is also a symmetric monoidal category.

Nice!

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).)

So, in the end it doesn't matter much. It does, however, illustrate the subtle difference between specifying things via a universal property (and thus 'up to canonical isomorphism') and specifying them 'on the nose' (that is, up to equality).