Thanks. I think my problem is the way I visualize the quantification: I think of two random arrows q and p (not necessarily composable), and the statement "q is a kernel of p" is like a binary predicate on the arrows of the category. So my above example goes through if I interpret "q is a kernel of p iff each q(n) is a kernel of p(n)" in my way

Sweet, didn't know there's such a simple example. To Tom Goodwillie and Martin Brandenburg, why do so many people use proper class? I follow MacLane's method and think of small being an element of a fixed universe U, and large as a set that's not in U. They are all "sets" in this way, and the class of all ordinals can be interpreted as the set of all small ordinals. If you insist on using proper class, how can you construct functor categories like Set^Set? My set theory is very weak but I'm pretty sure talking about Set^Set is like talking about the power class of the class of all sets.