I am reading a paper concerning the action of monoidal category to another category. Let $\[k\]$$k$ be a commutative ring, $\[R\]$$R$ is a k-algebra. $\[A=R-mod\]$$A=R-mod$, $\[B=R^{e}-mod=R\bigotimes _{k}R^{o}-mod\]$$B=R^{e}-mod=R\bigotimes _{k}R^{o}-mod$.
Consider the action:
$\[B\times A\rightarrow A,(M,N)\mapsto M\bigotimes _{R}N\]$$B\times A\rightarrow A,(M,N)\mapsto M\bigotimes _{R}N$ is an action of monoidal category of $\[R^{e}-mod=B^{~}=(B,\bigotimes _{R},R)\]$$R^{e}-mod=B^{~}=(B,\bigotimes _{R},R)$ on A.
The paper said this action induces the action
$\[\Phi : D^{-}(B)\times D^{-}(A)\to D^{-}(A)\]$$\Phi : D^{-}(B)\times D^{-}(A)\to D^{-}(A)$ of the monoidal derived category $\[D^{-}(B)\]$$D^{-}(B)$ on $\[D^{-}(A)\]$$D^{-}(A)$
I know this action should be $[(M,N)\mapsto M\bigotimes_{R}^{L}N\]$$(M,N)\mapsto M\bigotimes_{R}^{L}N$.
But I do not know how is this action of monoidal derived category on the other derived category induced by the action of monoidal abelian category. Is there a canonical way(A natural transformation)to get this action?
Notice that the action of monoidal abelian category is defined as follows
$\[\Psi:=(\Phi ,\phi ,\phi _{0})\]$$\Psi:=(\Phi ,\phi ,\phi _{0})$
$\[\Phi :B=(B,\bigotimes _{R},R)\rightarrow End(A)\]$$\Phi :B=(B,\bigotimes _{R},R)\rightarrow End(A)$
$\[\Phi (V)\cdot \Phi (W)\overset{\phi }{\rightarrow}\Phi (V\bigotimes _{R}W)\]$$\Phi (V)\cdot \Phi (W)\overset{\phi }{\rightarrow}\Phi (V\bigotimes _{R}W)$
The back ground of this question is localization of differential operator in derived category, so I added the tag"algebraic geometry"
This paper is "Differential Calculus in Noncommutative algebraic geometry I" which is available in MPIM