  We present a study of the notion of coalgebraic simulation introduced by
Hughes and Jacobs. Although in their original paper they allow any functorial
order in their definition of coalgebraic simulation, for the simulation
relations to have good properties they focus their attention on functors with
orders which are strongly stable. This guarantees a so-called
"composition-preserving" property from which all the desired good properties
follow. We have noticed that the notion of strong stability not only ensures
such good properties but also "distinguishes the direction" of the simulation.
For example, the classic notion of simulation for labeled transition systems,
the relation "p is simulated by q", can be defined as a coalgebraic simulation
relation by means of a strongly stable order, whereas the opposite relation, "p
simulates q", cannot. Our study was motivated by some interesting classes of
simulations that illustrate the application of these results:
covariant-contravariant simulations and conformance simulations.
