e have the following commutative diagrams of Grothendieck transformations:  0.  $$\xymatrix{     &   \bK_0(\m V_k^{qp} / - )  \ar [dl]_{mC_0} \ar [dr]^{{T_{0}}} \\     {\bK_{alg}( - ) } \ar [rr] _{\tau}& &  \bH(-) \otimes \bQ.}$$  0.  $$\xymatrix{     &   \bK_0(\m V_k^{qp} / - )  \ar [dl]_{\epsilon} \ar [dr]^{{T_{-0}}} \\     {\tilde{\bF}( - ) } \ar [rr] _{\gamma}& &  CH(-) \otimes \bQ,}$$ if $k$ is of characteristic zero. Here $\epsilon$ is the unique Grothendieck transformation satisfying the normalization condition $\epsilon\Bigl(\Bigl[[X  \xrightarrow{\op {id}_X}  X]\Bigr]\Bigr)=\jeden_f$ for a smooth morphism $f: X \to Y$. And similarly for the bivariant Chern class transformation $\gamma: \bF( - ) \to A^{PI}( - )\otimes \bQ \supset  CH(-) \otimes \bQ$ in case $k=\bC$.  0.  Assume $k$ is of characteristic zero. Then the associated covariant transformations in Theorem \[thm:main