Lemma 22.33.2. In the situation above, the left derived functor of $F$ exists. We denote it $- \otimes _ A^\mathbf {L} N : D(A, \text{d}) \to D(B, \text{d})$.

**Proof.**
We will use Derived Categories, Lemma 13.14.15 to prove this. As our collection $\mathcal{P}$ of objects we will use the objects with property (P). Property (1) was shown in Lemma 22.20.4. Property (2) holds because if $s : P \to P'$ is a quasi-isomorphism of modules with property (P), then $s$ is a homotopy equivalence by Lemma 22.22.3.
$\square$

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