{\em Galois comodules} over a coring can be characterised by properties of
the relative injective comodules. They motivated the definition of {\em Galois
functors} over some comonad (or monad) on any category and in the first section
of the present paper we investigate the role of the relative injectives
(projectives) in this context.
Then we generalise the notion of corings (derived from an entwining of an
algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a
key role is played by the notion of a {\em grouplike natural transformation}
g:I→G generalising the grouplike elements in corings. We apply the
evolving theory to Hopf monads on arbitrary categories, and to comonoidal
functors on monoidal categories in the sense of A. Brugui\`{e}res and A.
Virelizier. As well-know, for any set G the product G×− defines an
endofunctor on the category of sets and this is a Hopf monad if and only if G
allows for a group structure. In the final section the elements of this case
are generalised to arbitrary categories with finite products leading to {\em
Galois objects} in the sense of Chase and Sweedler