The notion of a comonadic functor is the dual of that of a monadic functor. See there for more background.

Definition

Given a pair $L\dashv R$ of adjoint functors, $L\colon A \to B\colon R$, with counit $\epsilon$ and unit $\eta$, one forms a comonad$\mathbf{\Omega} = (\Omega, \delta, \epsilon)$ by $\Omega \coloneqq L \circ R$, $\delta \coloneqq L \eta R$. $\mathbf{\Omega}$-comodules (aka $\mathbf{\Omega}$-coalgebras) form a category $B_{\mathbf{\Omega}}$ and there is a natural comparison functor $K = K_{\mathbf{\Omega}}\colon A \to B_{\mathbf{\Omega}}$ given by $A \mapsto (L A, L A \stackrel{L(\eta_A)}\to L R L A)$.

A functor $L\colon A\to B$ is comonadic if it has a right adjoint $R$ and the corresponding comparison functor $K$ is an equivalence of categories. The adjunction $L \dashv R$ is said to be a comonadic adjunction.

Properties

Beck’s monadicity theorem has its dual, comonadic analogue. To discuss it, observe that for every $\Omega$-comodule $(N, \rho)$,

If $T: Set \to Set$ is a monad on $Set$, with corresponding monadic functor $U: Set^T \to Set$, then the left adjoint $F: Set \to Set^T$ is comonadic provided that $F(!): F(0) \to F(1)$ is a regular monomorphism and not an isomorphism. In particular, if $T$ is given by an algebraic theory with at least one constant symbol and at least one function symbol of arity greater than zero, then the left adjoint is comonadic (because then $F(!): F(0) \to F(1)$ is a split monomorphism but not an isomorphism).

More generally, let $A$ be a locally small category with small copowers, and suppose $a$ is an object such that $0 \to a$ is a regular monomorphism but not an isomorphism, then the copowering with $a$,

$- \cdot a: Set \to A$

(left adjoint to $A(a, -): A \to Set$) is comonadic. See proposition 6.13 and related results in this paper by Mesablishvili.