  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.
