ag.algebraic geometry - How is this action of monoidal derived category induced?

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. $A=R-mod$, $B=R^{e}-mod=Rbigotimes _{k}R^{o}-mod$.

Consider the action:

$Btimes Arightarrow A,(M,N)mapsto Mbigotimes _{R}N$ 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 $(M,N)mapsto Mbigotimes_{R}^{L}N$.

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

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

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

$Phi (V)cdot Phi (W)overset{phi }{rightarrow}Phi (Vbigotimes _{R}W)$

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