 Over the past two decades, different proposals have been made to formulate a theory of mathematical structuralism based on category theory, thus a theory, or theories, of “categorical structuralism”.  We are now in a better position to consider these proposals, although we will still proceed indirectly, starting with more background. Category theory was first introduced as a branch of abstract algebra in Eilenberg &amp; Mac Lane’s famous article, “General Theory of Natural Equivalences” (1945). It subsequently developed into an autonomous mathematical discipline, in work by Mac Lane, Grothendieck, Kan, Lawvere, and many others, with important, wide-ranging applications in algebraic topology and homological algebra and, more recently, in computer science and logic (cf. Landry &amp; Marquis 2005, also the  entry on category theory in this encyclopedia).