Let $A,B,C$ be sheaves of (possibly noncommutative) rings, $M$ a $(B,A)$-bimodule and $N$ a $(A,C)$-bimodule.
The object I am trying to understand is $M\otimes_{A}^{L}N$. I can choose a resolution $S^{\cdot}$ of $M$ conisting of flat right-$A$-modules and I can also choose a resolution $T^{\cdot}$ of $N$ consisting of flat left-$A$-modules and both $S^{\cdot}\otimes_{A}N$ and $M\otimes_{A}T^{\cdot}$ will be quasi isomorphic to $M\otimes_{A}^{L}N$ in the derived category of sheaves of abelian groups. While $S^{\cdot}\otimes_{A}N$ even represents a complex of right $C$-modules and $M\otimes_{A}T^{\cdot}$ represents a complex of left $B$-modules, I was wondering if it was possible to consider $\_\otimes_{A}^{L}\_$ as a functor $D^{-}(Mod(B,A))\times D^{-}(Mod(A,C))\rightarrow D^{-}(Mod(A,C))$ and if this is possible, how to compute $M\otimes_{A}^{L}N$.
I guess it should be possible to find a resolution of $M$ consisting of modules which are flat both as left- $B$- and as right $A$- modules, but even if I do: The computation of $M\otimes_{A}^{L}N$, considered as an object in the derived category of abelian sheaves does not depend on the choice of acyclic resolution with right $A$-modules, but is it also independent from this choice in $D^{-}(Mod(B))$, the derived category of left-$B$-modules bounded below?
I'd be happy about both ideas and references.