In this situation, an important feature of Galois connections is that one adjoint uniquely determines the other. Hence one can strengthen the above statement to guarantee that any supremum-preserving map between complete lattices is the lower adjoint of a unique Galois connection. The main property to derive this uniqueness is the following: For every in , is the least element of such that . Dually, for every in , is the greatest in such that . The existence of a certain Galois connection now implies the existence of the respective least or greatest elements, no matter whether the corresponding posets satisfy any completeness properties. Thus, when one upper adjoint of a Galois connection is given, the other upper adjoint can be defined via this same property.