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.

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.