0PD6 Lemma 8.2.8. The map qq defines a morphism of (𝒮M∙(Z),𝒮M∙(Z))({\mathcal{S}}_{M}^{\bullet}(Z),{\mathcal{S}}_{M}^{\bullet}(Z))-bimodules G→Id𝒮M∙(Zξ)G\to\operatorname{Id}\nolimits_{{\mathcal{S}}_{M}^{\bullet}(Z_{\xi})} and q(α∗α′)=q(α)⋅q(α′)q(\alpha\ast\alpha^{\prime})=q(\alpha)\cdot q(\alpha^{\prime}). Given h∈Hn∙h\in H^{\bullet}_{n} and α∈Gn\alpha\in G_{n}, we have q(hα)=q(αh)q(h\alpha)=q(\alpha h).