On monotone modalities and adjointness

We fix a logical connection (Stone ˧ Pred : Setop → 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 ω. We then consider monotone unary modalities arising from the logical connection and show that they all are left (or right) adjoints relative to ω