Lemma 24.28.1. In the situation above, the functor (24.28.0.1) composed with the localization functor $K(\textit{Mod}(\mathcal{A}', \text{d})) \to D(\mathcal{A}', \text{d})$ has a left derived extension $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}', \text{d})$ whose value on a good right differential graded $\mathcal{B}$-module $\mathcal{P}$ is $f^*\mathcal{P} \otimes _\mathcal {A} \mathcal{N}$.
Proof. Recall that for any (right) differential graded $\mathcal{B}$-module $\mathcal{M}$ there exists a quasi-isomorphism $\mathcal{P} \to \mathcal{M}$ with $\mathcal{P}$ a good differential graded $\mathcal{B}$-module. See Lemma 24.23.7. Hence by Derived Categories, Lemma 13.14.15 it suffices to show that given a quasi-isomorphism $\mathcal{P} \to \mathcal{P}'$ of good differential graded $\mathcal{B}$-modules the induced map
\[ f^*\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \longrightarrow f^*\mathcal{P}' \otimes _\mathcal {A} \mathcal{N} \]
is a quasi-isomorphism. The cone $\mathcal{P}''$ on $\mathcal{P} \to \mathcal{P}'$ is a good differential graded $\mathcal{A}$-module by Lemma 24.23.2. Since we have a distinguished triangle
\[ \mathcal{P} \to \mathcal{P}' \to \mathcal{P}'' \to \mathcal{P}[1] \]
in $K(\textit{Mod}(\mathcal{B}, \text{d}))$ we obtain a distinguished triangle
\[ f^*\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to f^*\mathcal{P}' \otimes _\mathcal {A} \mathcal{N} \to f^*\mathcal{P}'' \otimes _\mathcal {A} \mathcal{N} \to f^*\mathcal{P}[1] \otimes _\mathcal {A} \mathcal{N} \]
in $K(\textit{Mod}(\mathcal{A}', \text{d}))$. By Lemma 24.23.8 the differential graded module $f^*\mathcal{P}'' \otimes _\mathcal {A} \mathcal{N}$ is acyclic and the proof is complete. $\square$
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