  Assuming the Hodge conjecture for abelian varieties of CM-type, one obtains a
good category of abelian motives over the algebraic closure of a finite field
and a reduction functor to it from the category of CM-motives. Consequentely,
one obtains a morphism of gerbes of fibre functors with certain properties. We
prove unconditionally that there exists a morphism of gerbes with these
properties, and we classify them.
