  A theory of a derivator version of six-functor-formalisms is developed, using
an extension of the notion of fibered multiderivator due to the author. Using
the language of (op)fibrations of 2-multicategories this has (like a usual
fibered multiderivator) a very neat definition. This definition not only
encodes all compatibilities among the six functors but also their interplay
with homotopy Kan extensions. One could say: a nine-functor-formalism. This is
essential for dealing with (co)descent questions. Finally, it is shown that
every fibered multiderivator (for example encoding any kind of derived
four-functor formalism $(f_*, f^*, \otimes, \mathcal{HOM})$ occurring in
nature) satisfying base-change and projection formula formally gives rise to
such derivator six-functor-formalism in which $f_!=f_*$, i.e. a derivator
Grothendieck context.
