I am reading a paper concerning the action of monoidal category to another category. Let $\[k\]$ be a commutative ring, $\[R\]$ is a k-algebra. $\[A=R-mod\]$, $\[B=R^{e}-mod=R\bigotimes _{k}R^{o}-mod\]$.

Consider the action:

$\[B\times A\rightarrow A,(M,N)\mapsto M\bigotimes _{R}N\]$ is an action of monoidal category of $\[R^{e}-mod=B^{~}=(B,\bigotimes _{R},R)\]$ on A.

The paper said this action induces the action

$\[\Phi : D^{-}(B)\times D^{-}(A)\to D^{-}(A)\]$ of the monoidal derived category $\[D^{-}(B)\]$ on $\[D^{-}(A)\]$

I know this action should be $[(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

$\[\Psi:=(\Phi ,\phi ,\phi _{0})\]$

$\[\Phi :B=(B,\bigotimes _{R},R)\rightarrow End(A)\]$

$\[\Phi (V)\cdot \Phi (W)\overset{\phi }{\rightarrow}\Phi (V\bigotimes _{R}W)\]$

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