  In this article, we continue the study of monadic distributive lattices (or
m-lattices) which are a natural generalization of monadic Heyting algebras,
introduced by Monteiro and Varsavsky and developed exhaustively by
Bezhanishvili. First, we extended the duality obtained by Cignoli for
Q-distributive lattices to m-lattices. This new duality allows us to describe
in a simple way the subdirectly irreducible algebras in this variety and in
particular, to characterize the finite ones. Next, we introduce the category
mKF whose objects are monadic augmented Kripke frames and whose morphisms are
increasing continuous functions verifying certain additional conditions and we
prove that it is equivalent to the one obtained above. Finally, we show that
the category of perfect augmented Kripke frames given by Bezhanishvili for
monadic Heyting algebras is a proper subcategory of mKF.
