The basic notions are in fact not very articulate in themselves, and throughout the history of mathematics it has taken further ideas to articulate them. Bill saw how to articulate these and many more, quite directly, in categorical terms not assuming any prior set theory. That articulation works even if you do not take it as foundational. But it gets a natural foundational character in the framework of the category of categories -- thus CCAF, the axiomatic theory of the category of categories as a foundation.
I agree with you about generalities concerning pre-formal and formal concepts. A reason why I say CCAF is not a satisfactory categorical foundation is different. ETC is the formal basis of CCAF and ETC relies on a pre-formal notion of set or collection just like ZF or any other axiomatic theory built with Hilbert-Tarski axiomatic method. Elements of a new properly categorical method of theory-building are present in the "basic theory" (BC) that follows ETC. (I mean, in particular, the "redefinition" of functor in BC as 2-->A, etc. The standard definition of functor given earlier in ETC never reappears in BC.) However in CCAF these new features are not yet developed into an autonomous axiomatic method - or into a new way of formalisation of pre-formal concepts, if you like. In my understanding, such a method should meake part of categorical foundations deserving the name. CCAF remains in this sense eclectic, it is a half-way to categorical foundations. best, andrei [For admin and other information see: http://www.mta.ca/~cat-dist/ ]