Graphs of tensoring modules Let $A$ be a ring, and $L,M,N$ an $A^\mathrm{op} \otimes A$-modules.
$L \otimes_A M$ is then an $A \otimes A^\mathrm{op}$ module so we can tensor it again with $N$ to get an abelian group $(L \otimes_A M) \otimes_{A \otimes A^\mathrm{op}} N$.
This is basically tensoring $L, M, N$ in a triangle, in a cyclic fashion. I am under the impression that this idea is useful in understanding Hochschild (co)homology, but I don't actually know anything about it unfortunately.
My question is that given that we could form tensor products of more complicated graphs than a cyclic triangle, even varying $A$, or having multimodules instead of bimodules, does "tensoring diagrams of multimodules" exist in the literature?
 A: Consider a bicategory $\mathrm{Mod}$ whose objects are rings $A$, $B$..., whose morphisms $M : A \rightarrow B$ are $A$-$B$ bimodules (which are the same as $B^\mathrm{op}\otimes A$-modules), and whose 2-cells are bimodule maps. Composite of bimodules $M : A \rightarrow B$ and $M : B \rightarrow C$ is defined by $N\circ M = N\otimes_B M$.
$\mathrm{Mod}$ is a monoidal bicategory. The tensor product of objects is the tensor product of rings. The monoidal unit $I$ is the ring of integers.
$A^\mathrm{op}$ is the bidual of $A$ in $\mathrm{Mod}$. The unit $I \rightarrow A^\mathrm{op}\otimes A$ and the counit $A^\mathrm{op}\otimes A \rightarrow I$ both are $A$ considered as a module.
Compositions in  $\mathrm{Mod}$ correspond to various tensorings of modules. While, to represent  compositions in  $\mathrm{Mod}$ one can use the graphical calculus for monoidal categories with duals. 
Thus, given modules $L, M, N : A \rightarrow A$, the tensor product in the OP can be graphically represented as:

More precisely, this string diagram represents the morphism $I \rightarrow I$ of $\mathrm{Mod}$ which is the tensor product in the OP. 
One can draw various such string diagrams involving modules, which will correspond to different kinds of tensorings. "Multimodules" can be represented by a labeled nod which has few string coming in and going out from it, labeled by rings, which can indeed vary.
