Coalgebraic structures have recently been paid a lot of
attention. On the one hand, they give a systematic and uniform
treatment to many interesting and important concepts in
computer science, such as objects (in the sense of object-oriented
programming), automata, labelled transition systems, and processes.
On the other hand, they give us a powerful proof technique called
coinduction, which is based on (generalised) bisimulation relations.
Coalgebras for polynomial functors, in particluar, give us
intuitive models of behavioural specifications.

If the functors are resticted to polynomials, however,
coalgebras themselves do not have enough expressive power
for many applications, and there have been a couple of proposals
to overcome the limitations. In one direction, more and more
general functors have been considered. One major drawback of this
approach is that it becomes more and more difficult to ensure
the existance of final coalgebras (hence bisimilarities).
Another solution is to introduce algebraic structures on top of
coalgebraic ones. Compared with the former, this approach is of
limited expressive power, but always ensures the existence of
bisimilarities, which validate proofs by coinduction.