I think this is more a failure of documentation than anything else. The definition provided there is correct and useful for a certain subset of people, but there are ways of grokking comonads and cofree comonads that don't require so much math.

It's a tricky concept no matter how you slice it (in my opinion), but I like the following article:

http://dlaing.org/cofun/posts/free_and_cofree.html

I won't claim it's easy reading, but it's a lot more grounded and motivated.