[ghancock]

> On the other hand, I've not seen a rigorous set of axioms for category theory that didn't presuppose a notion of set or category or "collection".

I never worked through the details, and I'm not a category theory partisan, but I think that's what this is: https://ncatlab.org/nlab/show/fully%20formal%20ETCS

[Twisol]

Yes, this is essentially what I'm getting at in my cousin comment -- although the nlab is typically rather impenetrable to non-specialists.