  We use dialgebras, generalising both algebras and coalgebras, as a complement
of the standard coalgebraic framework, aimed at describing the semantics of an
interactive system by the means of reaction rules. In this model, interaction
is built-in, and semantic equivalence arises from it, instead of being
determined by a (possibly difficult) understanding of the side effects of a
component in isolation. Behavioural equivalence in dialgebras is determined by
how a given process interacts with the others, and the obtained observations.
We develop a technique to inter-define categories of dialgebras of different
functors, that in particular permits us to compare a standard coalgebraic
semantics and its dialgebraic counterpart. We exemplify the framework using the
CCS and the pi-calculus. Remarkably, the dialgebra giving semantics to the
pi-calculus does not require the use of presheaf categories.
