The important missing context is that a monad has `(return, bind)`, while a comonad has `(extract, extend)`, and both have their own respective laws letting you reason about combinations of those operations.

You can certainly do what you describe if you have a monad, but since you can't even try to unwrap a value if you don't know what the monad is concretely, you have no way to write a function `T a -> b` that's generic over T. Functions `T a -> b` only really make sense for comonads, which give you the `extract` and `extend` functions as primitive.