 Beyond the issue of methodological structuralism, the literature on categorical structuralism during the last 20 years centers around two questions that have already been mentioned. First, in what sense does category theory provide a framework for philosophical structuralism? Second, why is it better suited for that task than other frameworks, such as set theory, Shapiro’s structure theory, and Hellman’s modal logic, as has been claimed?  In recent work on these topics, one can find three related philosophical assumptions that characterize categorical structuralism and distinguish it from the versions of structuralism surveyed earlier. We will treat each of them in turn, starting again with Awodey’s writings.