TITLE: Motivic cohomology is universal with respect to what (co)homology theories?
QUESTION [9 upvotes]: I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem?
More generally, what can be said if one allows general schemes over a sufficiently good base $S$ (probably something like geometrically unibranch?) or change the topology (e.g., étale motivic cohomology)? 
Still more generally, what can be said if one uses Fulton-Macpherson's bivariant theories or Mixed Weil cohomology theories?

REPLY [5 votes]: I think a proof that motivic Borel-Moore homology is a universal Borel-Moore homology theory could be deduced from work of Bloch, Geisser-Levine, and Voevodsky, at least for varieties over fields. I have never seen the details of this written down.
Bloch sketches the fact that higher Chow groups have a cycle class map to certain cohomology theories in "Algebraic cycles and the Beilinson conjectures." A paper of Geisser-Levine elaborate on this, in the case of etale cohomology. I think that these methods can be used to show that higher Chow groups are the universal Borel-Moore homology theory.
Voevodsky proves that motivic Borel-Moore homology for smooth varieties over a field are isomorphic to higher Chow groups. See "Lectures on Motivic Cohomology" by Mazza-Weibel-Voevodsky. It should be possible to verify that this isomorphism is induced by the map from higher Chow groups constructed via the method of Bloch and Geisser-Levine.