On 18 Nov 2015, at 14:51, Vladimir Voevodsky wrote:

Hello, could anyone point me out to a proof that the category F whose objects are natural numbers and morphisms are morphisms of finite sets is a free category with coproducts where the associativity and unity isomorphisms are identities generated by one object? Thank you in advance, Vladimir.

In this paper M. Grandis, Finite sets and symmetric simplicial sets, Theory Appl. Categ. 8 (2001), No. 8, 244-252 the following results are proved, and point (a') should be related to what you want: The category of finite cardinals, equivalent to the category of finite sets and the site of augmented symmetric simplicial sets, is: (a') the free strict monoidal category with an assigned symmetric monoid; (b') the subcategory of Set generated by finite cardinals, their faces, degeneracies and main transpositions; (c') the category generated by faces, degeneracies and main transpositions, under the symmetric cosimplicial relations. The properties above are related to well-known (see Mac Lane) characterisations of the category of finite ordinals, the site of augmented simplicial sets: (a) the free strict monoidal category with an assigned internal monoid; (b) the subcategory of Set generated by finite ordinals, their faces and degeneracies; (c) the category generated by faces and degeneracies, under the cosimplicial relations. Regards, Marco [For admin and other information see: http://www.mta.ca/~cat-dist/ ]