Abstract: It is increasingly popular for philosophers of science to use category theory, the mathematical theory of structure, to adjudicate debates about the (in)equivalence of formal physical theories. In this talk, I discuss the theoretical foundations of this strategy. I introduce the concept of a “representation diagram" as a way to scaffold narrative accounts of how mathematical gadgets represent target systems, and demonstrate how their content can be effectively summarised by what I call a “structure category". I argue that the narrative accounts contain the real content of an act of physical representation, and the category theoretic methodology serves only to make that content precise and conducive to further analysis. In particular, one can use tools from category theory to assess whether one physical formalism thus presented has more "properties", "structure", or "stuff" than another.