Patch Locale of a Spectral Locale in Univalent Type Theory

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, locally small, and small complete frames with small bases. This turns out to be nontrivial and involves predicative reformulations of several fundamental concepts of locale theory.