3. Existing Approaches

The kinds of foundational story provided by category theorists seem to me largely irrelevant to the problems which they identify
with set theoretic foundations.
So far as I understand them there are two kinds of problem.
The first kind is explained in the introduction to Saunders Mac Lane's "Categories for the Working Mathematician", in which
observes that many of the things which one would like to be categories aren't.
It should be fairly obvious here that the complaint arises from the well-foundedness of classical set theory, and is essentially
*ontological*.
The only answer to this kind of problem is a "non-well-founded" foundation system (set-theoretic or otherwise), and I have
not come across any category theorists arguing for this kind of system.

The other kind of problem is linguistic.
Category theory is often thought of as a better way of talking, and in this respect is somewhat coy.
Not only does category theory provide a new way of talking, it regards the previous idioms as disreputable, and therefore
a category theorist might seek a foundation system which has the same domain of discourse but a new vocabulary.
Perhaps the most common approach to categorical foundations follows this line, usually through topos theory.
The idea is not that sets are the wrong thing to have in your universe, but that we should talk about them in a more abstract
way.

The other kind of foundational innovation which has been advocated by category theorists (notably Mac Lane) is the use of
weak set theories.
This apparently on the grounds that the strength of existing theories is not necessary.
I am completely unaware of any category theoretic rationale for this kind of innovation, which is in any case, not very innovative,
and does not offer to fix either of the kinds of problem which I identified above (and I don't know of any kind of problem
at all that it fixes, except perhaps ontological vertigo).