The Stacks project

Proof. The statement means that we have

bifunctorially in $M$ and $L^\bullet $. To see this we may replace $M$ by a differential graded $E$-module $P$ with property (P). We also may replace $L^\bullet $ by a K-injective complex of $\mathcal{O}$-modules $I^\bullet $. The computation of the derived functors given in the lemmas referenced in the statement combined with Lemma 22.22.3 translates the above into

where $\mathcal{B} = \text{Comp}^{dg}(\mathcal{O})$. There is an evaluation map from right to left functorial in $P$ and $I^\bullet $ (details omitted). Choose a filtration $F_\bullet $ on $P$ as in the definition of property (P). By Lemma 22.20.1 and the fact that both sides of the equation are homological functors in $P$ on $K(\text{Mod}_{(E, \text{d})})$ we reduce to the case where $P$ is replaced by the differential graded $E$-module $\bigoplus F_ iP$. Since both sides turn direct sums in the variable $P$ into direct products we reduce to the case where $P$ is one of the differential graded $E$-modules $F_ iP$. Since each $F_ iP$ has a finite filtration (given by admissible monomorphisms) whose graded pieces are graded projective $E$-modules we reduce to the case where $P$ is a graded projective $E$-module. In this case we clearly have

as graded $\mathbf{Z}$-modules (because this statement reduces to the case $P = E[k]$ where it is obvious). As the isomorphism is compatible with differentials we conclude. $\square$