The Stacks project

Lemma 24.28.3. In Lemma 24.28.1 the functor $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}', \text{d})$ is equal to $\mathcal{M} \mapsto Lf^*\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}$.

Proof. Immediate from the fact that we can compute these functors by representing objects by good differential graded modules and because $f^*\mathcal{P}$ is a good differential graded $\mathcal{A}$-module if $\mathcal{P}$ is a good differential graded $\mathcal{B}$-module. $\square$

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