  Ultrafilters are useful mathematical objects having applications in
nonstandard analysis, Ramsey theory, Boolean algebra, topology, and other areas
of mathematics. In this note, we provide a categorical construction of
ultrafilters in terms of the inverse limit of an inverse family of finite
partitions; this is an elementary and intuitive presentation of a consequence
of the profiniteness of Stone spaces. We then apply this construction to answer
a question of Rosinger posed in arXiv:0709.0084v2 in the negative.
