The coalgebraic structure of Set-coalgebras is determined by properties
of their type functors. In particular, preservation of certain limits
is reflected by structural properties of the corresponding coalgebras.
After a survey of old and new results in this direction, we consider
decompositions of Set-functors which preserve the relevant properties.
We shall develop an intuitive framework, which sheds some new light
on Set-functors, on their role as coalgebraic types and on the semantics
of coalgebraic modalities.