Abstract
We fix a logical connection between categories Set and BA given by 2 as a schizophrenic object, and study coalgebraic modal logic that is induced by a functor T: Set --) Set that is finitary and standard and preserves weak pullbacks and finite sets. We prove that for any such T, the cover modality nabla is a left (and its dual delta is a right) adjoint relative to the finitary power set functor.
We then consider monotone unary modalities arising from the logical connection and show that they all are left (or right) adjoints relative to the finitary power set functor.