In category theory, you often run into what is known as "size" issues. That is, you run into the issue that the categories you try to define are too "big" to be sets, and so you need to use classes or even larger collections. There are a couple solutions, but one of them would be to use NF or NFU. Since they have a set of all sets, you do not need to worry about your categories being to big. Having a category of all categories which is an object of itself is no problem, for example.

My question is, are there any issues with NF or NFU that would make them problematic as a foundation for category theory. If so, what are these issues?

wantsize issues - it is a well-known theorem that if you have a small complete category, then it must be a preorder. So if you want to have a sensible notion of complete categories, you need to have a hierarchy of sizes, and require big categories to have all small limits (as opposed to big categories having all big limits or small categories having all small limits). That said, I haven't checked if Freyd's proof goes through in NF. It seems to depend on how exactly you write things down. $\endgroup$ – Dexter Chua Sep 5 '17 at 1:39a prioripreclude there also being an interesting version of category theory that is formulated in a foundational theory in which size distinctions are not similarly mandatory. (In particular, note that Freyd's theorem also fails in intuitionistic logic, and there are models with nontrivial complete small categories.) $\endgroup$ – Mike Shulman Sep 5 '17 at 10:39detailsabout this particular "NF does not prove the existence..." statement? If that's true (which I don't doubt), then I can discard NF for my purposes. $\endgroup$ – Peter Heinig Sep 16 '17 at 17:31