  Game comonads have brought forth a new approach to studying finite model
theory categorically. By representing model comparison games semantically as
comonads, they allow important logical and combinatorial properties to be
exressed in terms of their Eilenberg-Moore coalgebras. As a result, a number of
results from finite model theory, such as preservation theorems and
homomorphism counting theorems, have been formalised and parameterised by
comonads, giving rise to new results simply by varying the comonad.
  In this paper we study the limits of the comonadic approach in the
combinatorial and homomorphism-counting aspect of the theory, regardless of
whether any model comparison games are involved. We show that any standard
graph parameter has a corresponding comonad, classifying the same class. This
comonad is constructed via a simple Kan extension formula, making it the
initial solution to this problem and, furthermore, automatically admitting a
homomorphism-counting theorem.
