  For a split reductive group $G$ over a finite extension $L$ of ${\mathbb
Q}_p$, and a parabolic subgroup $P \subset G$ we examine functorial properties
of the functors ${\mathcal F}^G_P$ introduced in \cite{OS2}. We discuss the
aspects of faithfulness, projective and injective objects, Ext-groups and some
kind of adjunction formulas.
