Mike Shulman wrote in part:
Even in a category of sets, I don't see why choice is necessary in order to complete a small subcategory under finite limits and obtain a small subcategory. It seems to me that what is needed is rather the axiom of collection, which implies that we can find some *set* of objects containing *at least one* limit for every finite diagram in the original small subcategory; and then we can iterate countably many times to obtain a small category which contains at least one limit for any finite diagram therein.
The axiom of collection guarantees only *some* appropriate set of objects, so you need to choose one. To iterate this countably many times, you might need dependent choice. Unless I'm not understanding what you're doing. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]