Let be a cover of by functors having a right adjoint, with unit and counit . Each pair defines a monad with underlying endofunctor and if admit products of objects, the whole cover defines a comonad on , induced by the adjunction . (Cf. gluing categories from localizations).
Denote the projections , , then . The unit defines the coaction which locally gives rise to and the counit locally gives .
Domain localization for a cover
and a functor. (I will present here a sketch following mainly Gabi’s improvements and notation.) Then defines the following local data
where . Then the composition
is the identity and the diagram
commutes. Such pairs are objects of a category which depends on the cover of and on .
Given a natural transformation , one defines the corresponding local data by whiskering:
The data, form a morphism in . By the definition that means that the diagram
of natural transformations commutes. One also considers the subcategories and of product preserving functors. The above correspondences do not require that are localizations.
To get to the inverse of the localizarion of funtors for a cover, we first consider in the case of comonadicity of the cover .
Starting with a datum one defines by and as the unique map simultaneously factorizing for all the composition
through the composition .
Similarly, given a morphism , one defines as the unique natural transformation such that for each the diagram
Next step is to compose the restriction of the above functor to product preserving data and functors with another functor . For the latter, we need the existence of the appropriate equalizers, as in the construction of the inverse of the comparison functor in the proof of the Beck comonadicity theorem. Namely, given one defines the value of on the -comodule in , by the equalizer of