I’ve been having fun exploring adjunctions lately and thinking about how we can take a monad apart and compose it the other way to get a comonad, and vice versa. Often I’ll find that a comonad counterpart of a given monad gives an interesting perspective on that monad, and ditto for a monad cousin to a given comonad.

Let’s take an example. There is a category of monoids Mon with monoids as objects and monoid homomorphisms as arrows between them. Then there is a functor f...