Invariants of structures

We give a categorification of the notion of a mathematical structure originally given by Bourbaki in their set theory textbook. We show that any isomorphism-invariant property of a finite structure can be computed by counting the number of isomorphic copies of small substructures it contains. Our main theorem in this direction is a generalization of the classical result of Hilbert about elementary symmetric polynomials generating the algebra of all symmetric polynomials. We also show that, for structures built from sets, the Yoneda functor extends to a canonical embedding of any such category of structures into an associated category of structures in the sense of classical model theory.