The Stacks project

Remark 22.37.7. Let $A, B, F, N$ be as in Proposition 22.37.6. It is not clear that $F$ and the functor $G(-) = - \otimes _ A^\mathbf {L} N$ are isomorphic. By construction there is an isomorphism $N = G(A) \to F(A)$ in $D(B, \text{d})$. It is straightforward to extend this to a functorial isomorphism $G(M) \to F(M)$ for $M$ is a differential graded $A$-module which is graded projective (e.g., a sum of shifts of $A$). Then one can conclude that $G(M) \cong F(M)$ when $M$ is a cone of a map between such modules. We don't know whether more is true in general.