I always find it a bit weird, when people compare set theory to category theory like this. When talking about set theory as a foundation for mathematics, I always think of Zermelo-Fraenkel[1] set theory (possibly with Choice), which is an axiomatic system, from which you can build a lot of maths (at some point one might want to introduce universes[2], but whatever). I'm not aware of a similar axiomatic system using category theory, are you?

[1] https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... [2] https://en.wikipedia.org/wiki/Grothendieck_universe