Lemma 15.85.1. Let $R \to S$ and $S \to S'$ be ring maps. The canonical map $\mathop{N\! L}\nolimits _{S/R} \otimes _ S^\mathbf {L} S' \to \mathop{N\! L}\nolimits _{S/R} \otimes _ S S'$ induces an isomorphism $\tau _{\geq -1}(\mathop{N\! L}\nolimits _{S/R} \otimes _ S^\mathbf {L} S') \to \mathop{N\! L}\nolimits _{S/R} \otimes _ S S'$ in $D(S')$. Similarly, given a presentation $\alpha $ of $S$ over $R$ the canonical map $\mathop{N\! L}\nolimits (\alpha ) \otimes _ S^\mathbf {L} S' \to \mathop{N\! L}\nolimits (\alpha ) \otimes _ S S'$ induces an isomorphism $\tau _{\geq -1}(\mathop{N\! L}\nolimits (\alpha ) \otimes _ S^\mathbf {L} S') \to \mathop{N\! L}\nolimits (\alpha ) \otimes _ S S'$ in $D(S')$.

**Proof.**
Special case of Lemma 15.84.6.
$\square$

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