[housecarpenter]

It may be worth explicitly noting that category theory and set theory are not mutually exclusive. You can use set theory as the foundation, and build category theory on that, and then use category theory to do all that interconnection of fields and whatnot. To me it seems like the most natural way to do it, because set theory is simple and intuitive (kind of), whereas category theory is complicated and extremely abstract.

(Of course there is a lot of stuff in set theory that is counter-intuitive when you get to infinite sets, but I still think that's better than what you have with categories where they're just abstract algebraic objects and there isn't any intuition in the first place.)