Let $A,B$ be rings and $h:A\rightarrow\text{End}(E)$, $k:B\rightarrow\text{End}(E)$ two ring homomorphisms. If $h_\alpha\circ k_\beta=k_\beta\circ h_\alpha$ for all $(\alpha,\beta)\in A\times B$, we say that the two (left or right–I have used left) module structures defined on $E$ by $h$ and $k$ are compatible.
Let $(A_\lambda)_{\lambda\in L}$,$(B_\mu)_{\mu\in M}$ be two families of rings. An $((A_\lambda)_{\lambda\in L},(B_\mu)_{\mu\in M})$-multimodule is a set $E$ with, for each $\lambda\in L$, a left $A_\lambda$module structure and, for each $\mu\in M$, a right $B_\mu$-module structure, all these module structures being compatible with one another.
Does the bolded portion of this definition imply compatibility amongst the $A_\lambda$ (resp. $B_\mu$) as well? For example does it imply that $A_\lambda$ and $A_{\lambda'}$ are compatible for $\lambda,\lambda'\in L$?