  In the present paper we extend the theory of sheaves on moment graphs due to
Braden-MacPherson and Fiebig to the context of an arbitrary oriented
equivariant cohomology h (e.g. to algebraic cobordism). We introduce and
investigate structure h-sheaves on double moment graphs to describe equivariant
oriented cohomology of products of flag varieties. We show that in the case of
a total flag variety X of Dynkin type A the space of global sections of the
double structure h-sheaf also describes the endomorphism ring of the
equivariant h-motive of X.
