In a monoidal category $(\mathcal C, \otimes)$ one can consider algebras (internal monoids) and bimodules over those algebras.
For which algebras $A, B$ can we classify all $(A, B)$-bimodules? What about $(A, A)$-bimodules? For which categories $\mathcal C$ can we classify these for all algebras in $\mathcal C$?