Let $f:A \rightarrow B$ be a weak equivalence of simplicial commutative rings. There is a Quillen pair $(-\otimes\_{A}B, f\_{\ast})$ which is an equivalence. In this situation, $(-\otimes\_{A}B, f\_{\ast})$ is an equivalence precisely if the unit map 


$$\eta\_{M} : M \rightarrow f\_{\ast}(M \otimes\_{A}B)$$


is a weak equivalence for all cofibrant $A$-modules $M$. I am having trouble seeing why this map is a weak equivalence. My guess is use the sequence of $A$-modules $\ker(f) \rightarrow A \rightarrow B$ to compute the homotopy groups of the sequence $\ker(\eta\_{M}) \rightarrow M \rightarrow f\_{\ast}(M\otimes\_{A}B)$. 


Is this the right idea, or is there some easier way to demonstrate the pair is a Quillen equivalence?



Quillen equivalent module categories
The counit map is cocontinuous in M, so using the fact
that any cofibrant object is a retract of a transfinite composition
of cobase changes of generating cofibrations of A-modules,
combined with the left properness of the model category of A-modules
and the fact that the left adjoint sends generating cofibrations to monomorphisms and the right adjoint preserves monomorphisms,
the problem boils down to showing the claim
for the case when M is the domain or codomain of a generating cofibration.
In this case, this amounts to M=A[n] or M=(A[n−1]←A[n]),
and in both cases the claim is trivial.


