Stateful Runners of Effectful Computations

What structure is required of a set so that computations in a given notion of computation can be run statefully with this set as the state set? For running nondeterministic computations statefully, a resolver structure is needed; for interactive I/O computations, a “responder-listener” structure is necessary; to be able to serve stateful computations, the set must carry the structure of a lens. We show that, in general, to be a stateful runner of computations for a monad corresponding to a Lawvere theory (defined as a set equipped with a monad morphism between the given monad and the state monad for this set) is the same as to be a comodel of the theory, i.e., a coalgebra of the corresponding comonad. We work out a number of instances of this observation and also compare runners to handlers.