  In the present work, a natural sequel to \cite{MaPi1}, we further discuss the
existence 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 of
the $2$-categorical data involved in the construction of categories of
institution-like structures. More precisely, we remark that the process used
for passing from rooms to institutions (\cite{Diac2}) can be extended, due to
its $2$-functoriality, to more general room-like and institution-like
structures in such a way that the aforementioned adjunctions are all seen to
arise from simpler adjunctions at the room-like level. Secondly, and mostly
independently, we provide some applications of such adjunctions to abstract
logics, mainly to the setting of propositional logics and filter pairs
(\cite{AMP1}); we also generalize the process of skolemization, a classical
device from predicate logic, to the institutional setting.
