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In the present work, a natural sequel to \cite{MaPi1}, we further discuss theexistence of adjunctions between categories of institutions and of$\pi$-institutions. This is done at both a foundational and an applied level.Firstly, we reformulate and conceptually clarify such adjunctions in terms ofthe $2$-categorical data involved in the construction of categories ofinstitution-like structures. More precisely, we remark that the process usedfor passing from rooms to institutions (\cite{Diac2}) can be extended, due toits $2$-functoriality, to more general room-like and institution-likestructures in such a way that the aforementioned adjunctions are all seen toarise from simpler adjunctions at the room-like level. Secondly, and mostlyindependently, we provide some applications of such adjunctions to abstractlogics, mainly to the setting of propositional logics and filter pairs(\cite{AMP1}); we also generalize the process of skolemization, a classicaldevice from predicate logic, to the institutional setting.

Connecting abstract logics and adjunctions in the theory of ($\pi$-)institutions: some theoretical remarks and applications