 As has been argued repeatedly in the literature along the lines just summarized, category theory provides a more natural framework than traditional set theory for mathematical or methodological structuralism in mathematics. But what about its prospects as a form of philosophical structuralism, i.e., an alternative to the theories of Resnik, Shapiro, Hellman, etc.?  We already mentioned that Awodey (1996) also presents it as such; but this has led to ongoing controversies.  Hellmann (2003) contains a first critical discussion of philosophical claims such as Awodey’s. That is to say, Hellman’s article raises several objections against the view that category theory provides an adequate framework for a structuralist account of mathematics in the philosophical sense. As we will see, these objections are closely related to the status of category theory as a foundational discipline.