nLab
comonad

A comonad (or cotriple) on a category $A$ is a monad on its dual category $A^{\mathrm{op}}$. Similarly a comonad in a 2-category $E$ is a comonoid in the monoidal category $E(X,X)$ for some object $X$. Every additive comonad on an abelian category $A$ induces an augmented simplicial endofunctor on $A$, what is the basis of comonadic homology?. Comodules (= coalgebras) over the comonad with underlying endofunctor $M_R\mapsto M_R{\otimes }_{R}S$ in $R$-$\mathrm{Mod}$ for the extension of rings $R\hookrightarrow S$ correspond to the descent data for that extension. Gluing of categories from localizations may also be formalized via comonads.

Distributive laws between a monad and a comonad are so-called mixed distributive laws; a special case has been rediscovered in physics under the name entwining structures (Brzeziński, Majid 1997). Their theory is often studied in the connection with the theory of comonads in the bicategory of rings, modules and morphisms of modules, that is corings. There is a homomorphism of bicategories from a bicategory of entwinings to a bicategory of corings (Škoda 2008), which is an analogue of the 2-functor $\mathrm{comp}$ (R. Street, Formal theory of monads, JPAA 1972) of strict 2-categories in the case of distributive laws of monads (recall also that a distributive law among monads corresponds to a monad in the 2-category of monads).