One can define the projector $\Pi_{\delta}$ using another canonical element $\sum_{x\in Cr(f)}[S_{x}]\otimes R_{\delta}([U_{x}])$, instead of $\sum_{x\in Cr(f)}R_{\delta}([S_{x}])\otimes R_{\delta}([U_{x}])$, as we did. The above Proposition holds for the new canonical element as well.