It's entirely possible I don't understand what I am talking about, but I know that ZFC stands as a good foundation for much of mathematics and that category theory stands as a good foundation for other areas. Apparently TG set theory allows one to construct category theory inside of set theory, but is the reverse true? Is it possible to use category theory to construct set theory?
As in, is there an axiomatic version of category theory that can not only address homological algebra and so on, but at the same time neatly contain ZFC's axioms as theorems?
See Rethinking Set Theory by Tom Leinster. He describes Lawvere's representation of ZFC in detail in a way that people only a little bit familiar with category theory can understand.