 Third, arguably the most important feature of categorical structuralism is that it verifies a version of the “structuralist thesis” (hinted at earlier). Recall here Benacerraf’s argument in his 1965 paper that numbers should not be identified with particular sets, but rather, with positions in an abstract structure. Benacerraf also emphasizes that only certain properties are relevant in arithmetic.  For him, these are the number-theoretic properties, such as “being prime” or “being even”, that can be defined in terms of the primitive relations and functions of the theory at issue.  The general structuralist thesis holds, then, that all (relevant) properties of the objects treated by a mathematical theory should be structural in a specified sense. (The question what “structural” means also came up earlier.)