The bimodule characterization of morphisms between quantum currents

Let $(Q,\beta_{Q,-})$ and $(X,\beta_{X,-})$ be objects of the Drinfeld center, and let $(Q,\beta)^{s,\mathsf{M},t}_{\mathfrak L,\mathfrak R}$ and $(X,\beta)^{s,\mathsf{M},t}_{\mathfrak L',\mathfrak R'}$ be the corresponding quantum-current sectors, with local Hilbert spaces $\cH_s$, $\cH_t$, and $\cH_\mathsf{M}$. For sufficiently large boundary regions $\mathfrak L,\mathfrak R,\mathfrak L',\mathfrak R'$ and middle region $\cH_\mathsf{M}$, consider bimodule maps for the left and right local operator algebras.

Bimodule characterization. The morphisms between $(Q,\beta_{Q,-})$ and $(X,\beta_{X,-})$ in the Drinfeld center are in bijection with the $\operatorname{End}(\cH_s)\ot\operatorname{End}(\cH_t)$-bimodule maps between the sectors $(Q,\beta)^{s,\mathsf{M},t}_{\mathfrak L,\mathfrak R}$ and $(X,\beta)^{s,\mathsf{M},t}_{\mathfrak L',\mathfrak R'}$ for large enough $\mathfrak L,\mathfrak R,\mathfrak L',\mathfrak R'$ and $\cH_\mathsf{M}$.

This proposes a purely symmetric-tensor characterization of morphisms between quantum currents, intended to make their category equivalent to the Drinfeld center. The preceding discussion motivates using bimodule maps over local operator algebras, but notes that this approach is expected to work for half-infinite currents and not work well for finitely long currents; the stated bijection remains unverified in the supplied text.