Thanks to all who wrote something on this question. It clarified mi ignorance: There is the category of finite sets, namely, the category of all those sets which happen to be finite. No need of more precision. But it is not small (or an element of the universe if you like). If you want small, then there are plenty of them, and anybody can use their FAVORITE one. But this is not usually done, it seems that the fact that the canonical one is “essentially small” is good enough to dismiss all possible problems. For example, people which consider the presheaf category Set^((Set_f)^op) (object classifier) often do as if Set_f were canonical and small. Now, if you work with a Grothendieck base topos “as if it were the category of sets”, you are forced to specify which small category of finite sets you are using, or not ?. Cheers e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]