This question is in some sense a continuation to this question: Derived Nakayama for complete modules

For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\colon \mathcal C\rightarrow \mathcal D$ be a map of chain complexes of derived $I$-complete $A$-modules. I am trying to apply the "derived Nakayama" to the mapping cone of $f$ to produce the following result:

Suppose that $f'\colon \mathcal C\otimes^{\mathbf L} A/I\rightarrow \mathcal D\otimes^{\mathbf L} A/I$ is a quasi-isomorphism. Then $f$ is a quasi-isomorphism.

To do this, I want to relate the mapping cone $cone(f')$ to the mapping cone of $cone(f)\otimes^{\mathbf L}A/I$ but I am at a loss on how to proceed here. Any tips or solutions are welcome.