Toby Bartels writes:

John Baez wrote in part:

...

Claim: FinSet is the free biCartesian category on nothing.

What is the justification for including in the term "biCartesian" that the products distribute over the coproducts? If you add that the Cartesian product is closed (which it is in FinSet), *then* you get this, of course. So FinSet is either the free biCartesian category where products distribute over coproducts on nothing, or else the free Cartesian closed coCartesian category on nothing. It would be nice to have a single term like "biCartesian" for either of these concepts, but I don't see the justification, especially since the concept isn't very symmetric.

These categories are often called distributive. For an introduction to them, and their relationship with extensive categories, see the paper: Aurelio Carboni, Stephen Lack, and R.F.C. Walters, Introduction to extensive and distributive categories, J. Pure Appl. Alg. 84(1993), 145-158. Steve Lack.