We prove that any endofunctor on a \emph{class-theoretic} category has a
\emph{final coalgebra}. Moreover, we characterize  functors on
\emph{set-theoretic} categories which are \emph{identical} on objects, and
functors which are \emph{constant} on objects.  

Keywords: categories of sets, partially defined endofunctors, identity
functor, constant functor, final coalgebra.