 A series of articles by McLarty represent the first line of response well (e.g., McLarty 2004, 2011, 2012). Roughly speaking, his reply to Hellman is the following: Whereas category theory and general topos theory did originate as algebraic theories and are, thus, not feasible as foundational systems, certain theories of particular categories and toposes have been introduced as alternative foundations. McLarty’s central examples are Lawvere’s axiomatizations of the category of categories and his “Elementary Theory of the Category of Sets” (ETCS).