  Earlier, Lunts and Rosenberg studied a notion of compatibility of
endofunctors with localization functors, with an application to the study of
differential operators on noncommutative rings and schemes. Another
compatibility -- of Ore localizations of an algebra with a comodule algebra
structure over a given bialgebra -- introduced in my earlier work -- is here
described also in categorical language, but the appropriate notion differs from
that of Lunts and Rosenberg, and it involves a specific kind of distributive
laws. Some basic facts about compatible localization follow from more general
functoriality properties of associating comonads or even actions of monoidal
categories to comodule algebras. We also introduce localization compatible
pairs of entwining structures.
