In the context of quantaloid-enriched categories, we rely essentially on the classifying property of presheaf categories to give a conceptual proof of a theorem due to Hohle: the double power monad and the composite power monad, on the category of quantaloid-enriched categories, are the same. Via the theory of distributive laws, we identify the algebras of this monad to be the completely codistributive complete categories, and the homomorphisms between such algebras are the bicontinuous functors. With these results we hope to contribute to the further development of a theory of Q-valued preorders (in the sense of Pu and Zhang).
['Isar Stubbe']
The double power monad is the composite power monad