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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?

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    @QiaochuYuan Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-22

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