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 both adjoints preserve monomorphisms, the problem boils down to showing the claim for the case when M is a 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.

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.