There is an equivalence between modules over rings and Beck modules (Abelian group objects in a slice category)
$$ \text{Mod}_R \cong \text{Ab}(\text{CRing}/R) $$
I think of this as similar to the relationship between presheafs and discrete fibrations.
$$ [C^{\text{op}}, \text{Set}] \cong \text{Dis}(C) $$
There is also an equivalence between profunctors and two sided discrete fibrations
$$ [C \times D^{\text{op}}, \text{Set}] \cong \text{Dis}(C, D) $$
I feel like there ought to be a similar kind of equivalence between bimodules and some sort of "discrete spans" or "Beck bimodule".
But I'm not really familiar with the language of modules over rings. I primarily learn about functors and type theory stuff.
I'm not sure this is the right way to put it. Something like the following doesn't quite feel right to me.
$$ \text{Mod}_{R \oplus S} \cong \text{Ab}(\text{CRing}/(R \oplus S)) $$
I'm not really sure if an equivalence between the 2-category of bimodules plus intertwiners and a 2-category of "discrete spans" or "Beck bimodule" makes sense.