0PDP Proof. Using (8.1.1) and (8.1.5), we have isomorphisms Lξ1+∙(T,−)∧Rξ2−∙(−,S)→∼colimr,s→∞Hom𝒮∙(Z)(−,T⊔{ξ1+(mr+)})∧Hom𝒮∙(Z)(S⊔{ξ2−(ms−)},−)→∼colimr,s→∞Hom𝒮∙(Z)(S⊔{ξ2−(ms−)},T⊔{ξ1+(mr+)})→∼Hom𝒮∙(Z)(S⊔{ξ2−(−1)},T⊔{ξ1+(1)}).L_{\xi_{1}^{+}}^{\bullet}(T,-)\wedge R_{\xi_{2}^{-}}^{\bullet}(-,S)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\\ \operatorname{colim}\nolimits_{r,s\to\infty}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(-,T\sqcup\{\xi_{1}^{+}(m^{+}_{r})\})\wedge\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\sqcup\{\xi_{2}^{-}(m^{-}_{s})\},-)\\ \mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{colim}\nolimits_{r,s\to\infty}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\sqcup\{\xi_{2}^{-}(m^{-}_{s})\},T\sqcup\{\xi_{1}^{+}(m^{+}_{r})\})\\ \mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\sqcup\{\xi_{2}^{-}(-1)\},T\sqcup\{\xi_{1}^{+}(1)\}). and the lemma follows. ∎