This is a naive question on non naive foundations. Consider the inclusion S_f C S of finite sets in sets. Is the category S_f closed under finite limits and at the same time small ? For example, there are a proper class of singletons, all finite. Thus a proper class of empty limits. Question, which is the small category of finite sets ?, which are its objects ?. A small site with finite limits for a topos would not be closed under finite limits ? etc etc But, more basic is the question above: How do you define the small category of finite sets ? Or only there are many small categories of finite sets ? You can not define a finite limit as being any universal cone because then you get a large category. Then how do you determine a small category with finite limits without choosing (vade retro !!) some of them. And if you choose, which ones ? The esqueleton is small but a different question !! e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]