[voxl]

Set theory is fine, see the (Stack Project)[https://stacks.math.columbia.edu/browse] which develops a ton of modern Category Theory on ZFC (Zermelo-Fraenkel Set Theory with the Axiom of Choice) alone.

Alternative foundations of mathematics (Set Theory, Category Theory, Type Theory, and all their variations) can all mutually interpret the other by just postulating sufficiently large universes. You don't pick or advocate one based off its ability to encode mathematics, but instead based on its ability to express your intention and ideas.

Really its no different from programming language preference in my book.