  Our concrete objective is to present both ordinary bisimulations and
probabilistic bisimulations in a common coalgebraic framework based on multiset
bisimulations. For that we show how to relate the underlying powerset and
probabilistic distributions functors with the multiset functor by means of
adequate natural transformations. This leads us to the general topic that we
investigate in the paper: a natural transformation from a functor F to another
G transforms F-bisimulations into G-bisimulations but, in general, it is not
possible to express G-bisimulations in terms of F-bisimulations. However, they
can be characterized by considering Hughes and Jacobs' notion of simulation,
taking as the order on the functor F the equivalence induced by the epi-mono
decomposition of the natural transformation relating F and G. We also consider
the case of alternating probabilistic systems where non-deterministic and
probabilistic choices are mixed, although only in a partial way, and extend all
these results to categorical simulations.
