A categorical account is given of the Hofmann–Mislove theorem, describing the Scott open filters on a frame. The account is stable under an order duality and so is shown to also cover Bunge and Funk's constructive description of the points of the lower power locale.

The categorical axioms offered are based on a representation theorem for dcpo homomorphism between frames in terms of certain natural transformations; this allows for a categorical account to be given of dcpo homomorphisms. This specializes to give a new categorical description of the upper and lower power locale constructions.