  The paper explores categorical interconnections between lattice-valued
Relational systems and algebras of Fitting's lattice-valued modal logic. We
define lattice-valued boolean systems, and then we study co-adjointness,
adjointness of functors. As a result, we get a duality for algebras of
lattice-valued logic. Following this duality results, we establish a duality
for algebras of lattice-valued modal logic
