  Stone locales together with continuous maps form a coreflective subcategory
of spectral locales and perfect maps. A proof in the internal language of an
elementary topos was previously given by the second-named author. This proof
can be easily translated to univalent type theory using resizing axioms. In
this work, we show how to achieve such a translation without resizing axioms,
by working with large and locally small frames with small bases. This requires
predicative reformulations of several fundamental concepts of locale theory in
predicative HoTT/UF, which we investigate systematically.
