Let $A$ be a (Noetherian) local ring with finitely generated maximal ideal $I$ and suppose that it is $I$-adic complete and let $f:\mathcal C \rightarrow \mathcal D$ be map of (bounded in a suitable sense) chain complexes of finitely generated, $I$-complete $A$-modules. Let $-\otimes^L-$ denote the derived tensor-product (as left derived functor).
Is it possible to apply the derived Nakayama Lemma in some way to show that $f$ is a quasi-isomorphism provided $f':\mathcal C \otimes^L A/I \rightarrow \mathcal D \otimes ^L A/I$ is a quasi-isomorphism?