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).

> **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).