Category theory and set theory are complementary to one another, not in competition. I think this 'debate' is a bit of academic controversialising rather than an actual difference. If you've done a bit of category theory, you will realize how important the category of sets is (for Yoneda's lemma, representability, existence of generators, etc).

Even if you completely buy into homotopy type theory as a foundation for ∞-categories and homotopy theory, the theory of sets reappears in other garb as the theory of 0-types. A theory of sets is too natural an idea to escape.

I just also want to note: If you write out the syntactic version of ETCS, you end up with something that is more or less equivalent to ZFC. The ETCC, on the other hand, is widely considered to be a dead-end.

From the nLab:

As pointed out by J. Isbell in 1967, one of Lawvere’s results (namely, the theorem on the ‘construction of categories by description’ on p.14) was mistaken, which left the axiomatics dangling with insufficient power to construct models for categories. Several ways to overcome these problems where suggested in the following but no system achieved univocal approval (cf. Blanc-Preller(1975), Blanc-Donnadieu(1976), Donnadieu(1975), McLarty(1991)).

As ETCC also lacked the simplicity of ETCS, it rarely played a role in the practice of category theory in the following and was soon eclipsed by topos theory in the attention of the research community that generally preferred to hedge their foundations with appeals to Gödel-Bernays set-theory or Grothendieck universes.

Category theory and set theory are complementary to one another, not in competition. I think this 'debate' is a bit of academic controversialising rather than an actual difference. If you've done a bit of category theory, you will realize how important the category of sets is (for Yoneda's lemma, representability, existence of generators, etc).

Even if you completely buy into homotopy type theory as a foundation for ∞-categories and homotopy theory, the theory of sets reappears in other garb as the theory of 0-types. A theory of sets is too natural an idea to escape.

I just also want to note: If you write out the syntactic version of ETCS, you end up with something that is more or less equivalent to ZFC. The ETCC, on the other hand, is widely considered to be a dead-end.

From the nLab:

As pointed out by J. Isbell in 1967, one of Lawvere’s results (namely, the theorem on the ‘construction of categories by description’ on p.14) was mistaken, which left the axiomatics dangling with insufficient power to construct models for categories. Several ways to overcome these problems where suggested in the following but no system achieved univocal approval (cf. Blanc-Preller(1975), Blanc-Donnadieu(1976), Donnadieu(1975), McLarty(1991)).

As ETCC also lacked the simplicity of ETCS, it rarely played a role in the practice of category theory in the following and was soon eclipsed by topos theory in the attention of the research community that generally preferred to hedge their foundations with appeals to Gödel-Bernays set-theory or Grothendieck universes.

Edit: Just to clarify, I think most mathematicians working in category theory, homotopy theory, algebraic geometry, etc. are more or less agnostic about foundations, as long as they are equivalent in strength to ZFC (or stronger with universes). There have been arguments for ETCS(+Whatever) as a 'better' foundation, but when you get into hairy set-theoretic issues (for example, see the Appendix to lecture 2 of Scholze's notes on condensed mathematics), we are just as likely to work with ZFC because setting up ordinals in ETCS is an added annoyance. I added this edit just to clarify that I am not a partisan of either approach and appreciate both (and am not interested in bringing up this old argument about Tom's paper that I linked!!!)

Category theory and set theory are complementary to one another, not in competition. I think this 'debate' is a bit of academic controversialising rather than an actual difference. If you've done a bit of category theory, you will realize how important the category of sets is (for Yoneda's lemma, representability, existence of generators, etc).

Even if you completely buy into homotopy type theory as a foundation for ∞-categories and homotopy theory, the theory of sets reappears in other garb as the theory of 0-types. A theory of sets is too natural an idea to escape.

I just also want to note: If you write out the syntactic version of ETCS, you end up with something that is more or less equivalent to ZFC. The ETCC, on the other hand, is widely considered to be a dead-end.

Category theory and set theory are complementary to one another, not in competition. I think this 'debate' is a bit of academic controversialising rather than an actual difference. If you've done a bit of category theory, you will realize how important the category of sets is (for Yoneda's lemma, representability, existence of generators, etc).

Even if you completely buy into homotopy type theory as a foundation for ∞-categories and homotopy theory, the theory of sets reappears in other garb as the theory of 0-types. A theory of sets is too natural an idea to escape.

I just also want to note: If you write out the syntactic version of ETCS, you end up with something that is more or less equivalent to ZFC. The ETCC, on the other hand, is widely considered to be a dead-end.

From the nLab:

As pointed out by J. Isbell in 1967, one of Lawvere’s results (namely, the theorem on the ‘construction of categories by description’ on p.14) was mistaken, which left the axiomatics dangling with insufficient power to construct models for categories. Several ways to overcome these problems where suggested in the following but no system achieved univocal approval (cf. Blanc-Preller(1975), Blanc-Donnadieu(1976), Donnadieu(1975), McLarty(1991)).

As ETCC also lacked the simplicity of ETCS, it rarely played a role in the practice of category theory in the following and was soon eclipsed by topos theory in the attention of the research community that generally preferred to hedge their foundations with appeals to Gödel-Bernays set-theory or Grothendieck universes.