TITLE: Why do categorical foundationalists want to escape set theory?
QUESTION [38 upvotes]: This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full.
I know that it's possible to formulate category theory within set theory while still being albe to construct the useful things one would want from category theory. So as far as I understand, all normal mathematics that involves category theory can be done as long as a little caution is taken. 
I also know that some people (categorical foundationalsists) would still like to formulate category theory without use of or reference to set theory. While I admit that I am curious about this for curiosity's sake, I'm not sure if there are any practical motivations for doing this. The only reason for wanting to separate category theory from set theory that I have read about is for the sake of `autonomy of category theory'.
So my question is twofold: What other reasons might categorical foundationalists have for separating category theory from set theory, and what practical purposes might it serve to do this?

REPLY [5 votes]: Todd Trimble is right: not all categoriests who are interested in foundations reject set theory. It seems to me that category and set theory are deeply interrelated. Categorical problems, as the existence of some adjoint functors depends on some completions of ZFC. For example the statement every subcategory closed under limits of a locally presentable category is reflective (that is, the inclusion functor has a left adjoint) is equivalent to a set theoretical statement called Vopenka principle (see J. Adamek, J. Rosicky, Locally presentable and accessible categories, London Math. Soc. Lect. Notes, Cambridge University Press, 1994). There are lots of such problems, which depend on extensions of ZFC, both in category theory but also in other mathematical areas which are formulated categorical e.g. the Whitehead problem in abelian group theory: Is it true that every abelian group which has no nontrivial extensions through $\mathbb Z$ is projective (free)?