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 the space of global sections of the double structure h-sheaf also describes endomorphism ring of the equivariant h-motive of X.