mw

the following is my preferred defintion of monad
a monad M requires the following three functions, along with their names (programmers, mathematicians):
a -> M a ("singleton" or "unit")
M M a -> M a ("flatten" or "multiplication")
(a -> b) -> (M a -> M b) ("map", no name its deifnition is given by the definition of unit)
such that, where m : M a and mmm : M M M a:
flatten (map singleton m) = m = flatten (singleton m)
flatten (flatten mmm) = flatten (map flatten mmm)
and calling them map, flatten, and singleton makes it easy to see how a lot of data structures are monads

the following is my preferred defintion of comonad, for you to read at the same time
a comonad W (becuase W is an upside down M) requires the following three functions, along with their names:
M a -> a ("extract")
M M a -> M a ("duplicate")
(a -> b) -> (M a -> M b) ("map", no name its deifnition is given by the definition of extract)
such that, where w : W a:
extract (map duplicate w) = w = extract (duplicate m)
duplicate (duplicate w) = duplicate (map duplicate w)

and you can see how 
-in the defintiions, all the arrows point the other way
-in the axioms, the corresponding words were filled in, and then the order of the functions was flipped