In the references I know, a symmetric monoidal category is defined as a monoidal category equipped with some additional data, so in particular the monoidal product is a functor $$ \mathcal{C}\times \mathcal{C}\to \mathcal{C}. $$ The word unbiased is sometimes used to mean that there are products $$ \mathcal{C}^n\to \mathcal{C} $$ but that is not quite what I mean here. Rather, I want a monoidal category to give me an assignment for every finite set $S$ $$ \mathcal{C}^S\to \mathcal{C} $$ and for everything to be described in terms of functors involving the category of (unordered) finite sets, maybe some weak kind of functor since associativity isn't strict.

- Does this definition exist? That is, is there some obstruction that is not obvious to me that prevents it from working?
- Is it written down clearly and cleanly somewhere? This seems like a natural alternative to the standard definition that could sometimes be useful to eliminate certain bookkeeping.
- Why isn't this viewpoint more common? Or is it common and I've just been reading the wrong books?

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