Michael Shulman wrote in part:

We can make finitely many choices without any axiom of choice. Thus, for any natural number n, by applying collection n times, we can find *some* n^th iterate of the "construction". (Formally, we prove this by induction on n.) Applying the axiom of collection again over the natural numbers, we obtain a set which contains at least one n^th iterate of the "construction" for every natural number n. Taking the union of this set, we should obtain a set of objects whose corresponding full subcategory contains at least one limit of every finite diagram therein.

OK, I buy that. The part where we take the union is the step that doesn't generalise to arbitrary applications of DC. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]