  It is known that the construction of the frame of ideals from a distributive
lattice induces a monad whose algebras are precisely the frames and frame
homomorphisms. Using the Fakir construction of an idempotent approximation of a
monad, we extend B. Jacobs' results on lax idempotent monads and show that the
sequence of monads and comonads generated by successive iterations of this
ideal functor on its algebras and coalgebras do not strictly lead to a new
category. We further extend this result and provide a new proof of the
equivalence between distributive lattices and coherent frames by showing that
when the first inductive step in the Fakir construction is the identity monad,
then the ambient category is equivalent to the free algebras.
