TITLE: What would be an interesting example of a Co-algebra with a base category other than Set?
QUESTION [5 upvotes]: In most or perhaps all the examples of a co-algebra that I have seen, the properties of sets as the base category was used, like the existence of products and co-product and Cartesian closeness. Does anyone have an example of a co-algebra and a system which makes use of more peculiar categorical properties?

REPLY [3 votes]: See this MO question and this MO question for several examples of coalgebras in categories other than $(\text{Set}, \times)$.