  We provide a characterisation of the category KH of compact Hausdorff spaces
and continuous maps by means of categorical properties only. To this aim we
introduce a notion of filtrality for coherent categories, relating certain
lattices of subobjects to their Boolean centers. Our main result reads as
follows: Up to equivalence, KH is the unique non-trivial well-pointed pretopos
which is filtral and admits all set-indexed copowers of its terminal object.
