If is a monad on , with corresponding monadic functor , then the left adjoint is comonadic provided that is a regular monomorphism and not an isomorphism. In particular, if 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 is a split monomorphism but not an isomorphism).
More generally, let be a locally small category with small copowers, and suppose is an object such that is a regular monomorphism but not an isomorphism, then the copowering with ,
(left adjoint to ) is comonadic. See proposition 6.13 and related results in this paper by Mesablishvili.