Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

linearity and compatibility conditions. In particular we refer to the study of thespecial class of tame herds, which yields a correspondence with Galois functors(generalizing the historical <strong>results</strong> we started with). Moreover, under the regularityassumption on the formal dual structure related to a herd, one can construct acoherd. Conversely, beginning with a coherd over a regular formal codual structure,a herd can be obtained. By applying twice this process, one can start with a formaldual structure and a herd, construct a formal codual structure and a coherd andthen compute also a new formal dual structure. Under the extra assumption that thestarting formal dual structure is also a Morita context, the final formal dual structurecomputed from a tame herd comes out to be closely related to the starting one. Weconsider a few cases in which the monads are in fact isomorphic. As an application,we develop some examples. The first is given by a right Galois comodule from whichwe derive the herd. Then we simplify the setting and we study the Schauenburg caseof A/k a faithfully flat Hopf-Galois extension with respect to H. In this example wecan compute the comonads associated to the herd and the coherd. The last exampleis a non trivial example of a coherd. It allows to compute the two monads associatedand the equivalence between the module categories with respect to these monads.Finally we investigate the bicategory of balanced bimodule functors which are oneof the most useful tools in this work and is inspired by the balanced bimodules inthe classical sense.We developed the portions of the theory of herds, resp. coherds, we found moresuitable for our purposes.In the first part we collect some well-known <strong>results</strong> including proofs. It is aboutequalizers and coequalizers, contractible equalizers and coequalizers and notationfor adjunctions.Then we concentrate, in the second section, the needed materials for the sequelabout monads. Similarly we do for comonads. At this point we also includethe Beck’s Tripleability Theorem and the generalized version which introduce theEilenberg-Moore comparison functor and the categories of modules and comodules.We reserve a short section to the notion of distributive laws and above all thecorrespondence between distributive laws and liftings of monads and comonads.The next section introduces the notion of pretorsor and herd bringing all thedetails and the <strong>results</strong> needed to prove the equivalence between herds and Galoisfunctors in the tame case. The same has been done for the dual case of copretorsorsand coherds.Later on a section dedicated to a new fundamental functor built from a herdand a coherd respectively and then the theorem relating the starting formal dualstructure and the one obtained after applying the two processes from a herd andfrom a coherd.One section is dedicated to the equivalence between the module categories obtainedfrom a copretorsor and to the equivalence between the comodule categoriesobtained from a pretorsor.The following section is a collection of the examples we provide in this work, aboutherds and coherds first and then applications of Beck’s Theorem and of its generalization.In particular, the example of a coherd was produced during some usefuldiscussions with T. Brzeziński. In the subsection dedicated to Galois comodules we3