I'm not very experienced with respect to Category Theory. So if this question makes no sense I'm sorry. At any rate here is my question: If the existence or non-existence of specific sets can be independent of set theory, then how can it be that the category Set is complete under small limits?

For example, suppose you have a small category A, with objects that are linearly ordered spaces X such that: X is without smallest or largest element, X has CCC, X is complete, and X is dense in itself. And as morphisms for A, you take order preserving bijections. Now, let F be the forgetful functor from A to the underlying set.

How exactly can you define the limit over F inside of Set?

Another example, would be considering some set sized collection of Whitehead groups, call it X. Now, consider X as a category equipped with homomorphisms as morphisms. And let G be the forgetful functor from X into Set.

How exactly can one define the limit over G inside Set?

Or even better, suppose that G is the identity functor from X into Grps. Then, what happens?

Am I correct in saying that the answer depends drastically on the set theoretic universe you pick? If so, how is this not a problem with category theory?