Let $k$ be a commutative ring. Let $R$ be a $k$-algebra. Weibel defines a notion of "relative" Tor in his intro to homological algebra book. For a right $R$-module $M$ and a left $R$-module $N$, he defines $\mathrm{Tor}_*^{R/k}(M,N)$. If you look at the concrete description of it, page 288, you will realize that actually $$ \mathrm{Tor}_*^{R/k}(M,N)=H_*B_*(M,R,N)$$
the homology of the two-sided bar construction chain complex.
Now fast forward to page 302: he proves that, for an $R$-bimodule $M$, $$HH_*(R,M)=\mathrm{Tor}_*^{R^e/k}(M,R)$$
where $HH$ denotes Hochschild homology.
Recall also that the Hochschild complex of $R$ and $M$ is actually isomorphic to $M\otimes_{R^e}B(A,A,A)$. Putting all of this together, we conclude that
$H_*B_*(M,R^e,R)=H_*(M\otimes_{R^e} B(R,R,R))$.
If you inspect the complexes, you will see that the are not isomorphic, really. My question is:
Can one find a concrete (zig-zag of) quasi-isomorphisms between $B_*(M,R^e,R)$ and $M\otimes_{R^e} B(R,R,R)$?
