I am learning D-modules recently, and my question might be technical. It arises from Lemma 2.6.13 in [Hotta-Takeuchi-Tanisaki](http://www.math.columbia.edu/~scautis/dmodules/hottaetal.pdf)'s book, which states that there exists a canonical isomorphism
$$
R\mathcal {H}om\_{D\_X}(M^\cdot, D\_X) \otimes^L\_{D\_X} N^\cdot \xrightarrow{\cong} R\mathcal {H}om\_{D\_X}(M^\cdot, N^\cdot)
$$
where $M^\cdot$ and $N^\cdot$ are coherent complexes of $D\_X$-modules. And I get stuck with its proof: how to find such a canonical morphism?


I tried to find this in Kashiwara-Schapira's book "Sheaves on Manifolds". What I found was the following: (page 112, eq.(2.6.11)) let $\mathcal S\to \mathcal R$ be a morphism of sheaves of rings on $X$ whose image is contained in the center of $\mathcal R$ and $\mathcal S$ is commutative, then we have
$$
R\mathcal {H}om\_{\mathcal R}(F^\cdot, G^\cdot) \otimes^L\_{\mathcal S} H^\cdot \xrightarrow{\cong} R\mathcal {H}om\_{\mathcal R}(F^\cdot, G^\cdot\otimes^L\_{\mathcal S} H^\cdot)
$$
which is very similar to the above but cannot really imply it, since $D\_X$ is not commutative. So, how to fix this gap? Moreover, I will appreciate it if you have any idea in understanding the above canonical isomorphism.



A canonical isomorphism in derived categories of D-modules
A map of sheaves is an isomorphism if and only if it's an isomorphism on stalks. Thus, it's enough to check this for the stalk of these sheaves, and so on can just check that for a non-commutative ring $R$ and two complexes of modules, you have $RHom(M,R)\otimes^{L}\_RN\cong RHom(M,N)$. Since modules have free resolutions, you can assume that $M\cong R^m$ and $N\cong R^n$ are free modules. Now the statement just says that $n\times m$ matrices are the tensor product of row vectors and column vectors over $R$. 


