We give a direct proof that between two toposes, F and E, bounded over a base topos S, adjunctions L -| R: Loc_F -> Loc_E over Loc_S are Frobenius if and only if R commutes with the double power locale monad and finite coproducts. The proof uses only certain categorical properties of the category of locales, Loc. This implies that between categories axiomatized to behave like categories of locales, it does not make a difference whether maps are defined as structure preserving adjunctions (i.e. those that commute with the double power monads) or Frobenius adjunctions.
Keywords: topos, locale, geometric morphism, Frobenius reciprocity, power monad
2010 MSC: 06D22, 18D35, 18B40, 22A22
Theory and Applications of Categories, Vol. 33, 2018, No. 17, pp 476-491.