Are there any known interesting F-(co)algebras where F isn't a $Set$ endofunctor? Every example I can think of deals with sets: an algebra of $X\mapsto 1+X$ for natural numbers, an algebra of $X\mapsto 1+A\times X$ for lists, a coalgebra of $X\mapsto A\times X$ for streams, etc. I'm not alone in this, Adámek writes
Although all important examples of application of coalgebra seem to concern coalgebras in Set, there are good reasons to develop the whole theory in an abstract category, e.g., [...]
in his "Introduction to coalgebra".
Surely there are interesting cases dealing with domains? Are there any dealing with the categories of topological spaces, of algebras, etc?