We study properties of functors on categories of sets (classes) together with set (class) functions. In particular,  we investigate the notion of \emph{inclusion preserving functor}, and we discuss various  \emph{monotonicity} and \emph{continuity properties} of set functors. As a consequence of these properties, we show that some classes of set operators do not admit \emph{functorial extensions}.
Then, starting from Aczel's \emph{Special Final Coalgebra Theorem}, we study the class of functors \emph{uniform on maps}, we present and discuss various examples of functors which are \emph{not} uniform on maps but still inclusion preserving, and we discuss simple characterization theorems of final coalgebras as \emph{fixpoints}. We present a number of conjectures and problems.

Keywords: category of sets, set functor, inclusion preserving functor,  $\kappa$-based functor,
$\kappa$-reachable functor, functor uniform on maps, final coalgebra.