 wellfoundedness
Sauders Mac Lane in his classic texbook "Categories for the Working Mathematician" has an eary section on "Foundations".
It is the purpose of that section to clarify the foundational context in which the rest of the book is to be understood, and
the terminology of "small" (which actually means though he doesn't put it this way, "of accessible rank") and "large" categories
(which have the smallest inaccessible rank) which is adopted.
Here he remarks on the problems arising from the fact that many collections of interest to category theorists (for example,
the collection of all groups) are not sets in ZFC.
This problem is a consequence of the wellfoundedness of the ontology of ZFC (though I doubt that Mac Lane says that).


 lack of abstraction
Category theory is arguably, in a certain sense of the word, the most abstract kind of mathematics.
Some category theorist, believing that the mathematics should be as it is seen through category theory, find fault with the
method used in the derivation of mathematics in set theory.
This method is usually considered to consist in "coding up" (or representing) the various mathematical structures as sets.
The study of mathematical structures should be done without reference to particular exemplars of the structure, the structure
should be studied "up to isomorphism", in abstraction from its instances.
 ontological extravagence
Some category theorists regard category theory as being constructive, think that appropriate foundations for mathematics should also be constructive, and are hostile particularly to the outer
reaches of set theoretic ontology as being the antithesis of constructive mathematics.
Such constructivist antipathy to set theory is of course, not exclusive to category theorists.

