Lemma 15.87.12. Let $A$ be a ring. Let $E \to D$ be a morphism of $D(\mathbf{N}, A)$. Let $(E_ n)$, resp. $(D_ n)$ be the system of objects of $D(A)$ associated to $E$, resp. $D$. If $(E_ n) \to (D_ n)$ is an isomorphism of pro-objects, then for every $K \in D(A)$ the corresponding map

$R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A E_ n) \longrightarrow R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A D_ n)$

in $D(A)$ is an isomorphism (notation as in Remark 15.87.10).

Proof. Follows from the definitions and Lemma 15.86.10. $\square$

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