We study comodule functors for comonads arising from mixed distributive laws.
Their Galois property is reformulated in terms of a (so-called) regular arrow
in Street's bicategory of comonads. Between categories possessing equalizers,
we introduce the notion of a regular adjunction. An equivalence is proven
between the category of pre-torsors over two regular adjunctions (NA,RA)
and (NB,RB) on one hand, and the category of regular comonad arrows
(RA,ξ) from some equalizer preserving comonad C to NBRB on
the other. This generalizes a known relationship between pre-torsors over equal
commutative rings and Galois objects of coalgebras.Developing a bi-Galois
theory of comonads, we show that a pre-torsor over regular adjunctions
determines also a second (equalizer preserving) comonad D and a
co-regular comonad arrow from D to NARA, such that the
comodule categories of C and D are equivalent.Comment: 34 pages LaTeX file. v2: a few typos correcte