 According to McLarty, these theories should be understood as assertoric in Hellman’s sense. That is to say, their axioms are not merely implicit definitions, but general existential claims about categories, sets, and functions. ETCS, for instance, presents a function-based set theory, where sets and mappings between them form a topos. In contrast to ZFC, with its primitive membership relation, in ETCS a set is not specified in terms of its internal composition, but rather in terms of its mapping properties with respect to other sets, which are formulated independently of ZFC. McLarty’s response to the two objections mentioned above is then that categorical set theories such as ETCS do provide a foundation for mathematics that is logically autonomous from traditional, non-structural set theories. Moreover, given that mathematical structures can only be encoded in ETCS as objects up to isomorphism, such categorical set theories provide a more adequate foundation for modern structural mathematics than ZFC.  (We will return to this point below.)