A class K of coalgebras for an endofunctor T on the category of sets is a behavioural covariety if it is closed under disjoint unions and images of bisimulation relations (hence closed under images and domains of coalgebraic morphisms, including subcoalgebras). K may be thought of as the class of all coalgebras that satisfy some computationally significant property. In any logical system suitable for specifying properties of state-transition systems in the Hennessy-Milner style, each formula will define a class of models that is a behavioural variety.

Assume that the forgetful functor on T-coalgebras has a right adjoint, providing for the construction of cofree coalgebras, and let G^T be the comonad arising from this adjunction. Then we show that behavioural covarieties K are (isomorphic to) the Eilenberg-Moore categories of coalgebras for certain comonads G^K naturally associated with G^T. These are called pure subcomonads of G^T, and a categorical characterization of them is given, involving a pullback condition on the naturality squares of a transformation from G^K to G^T.

We show that  there is a bijective correspondence between behavioural covarieties of T-coalgebras and isomorphism classes of pure subcomonads of G^T.