  For a variety X which admits a Cox ring we introduce a functor from the
category of quasi-coherent sheaves on $X$ to the category of graded modules
over the homogeneous coordinate ring of $X$. We show that this functor is
right-adjoint to the sheafification functor and therefore left-exact. Moreover,
we show that this functor preserves torsion-freeness and reflexivity. For the
case of toric sheaves we give a combinatorial characterization of its
right-derived functors in terms of certain right-derived limit functors.
