A planar algebraic description of conditional expectations

Let $\mathcal{N}\subset\mathcal{M}$ be a unital inclusion of arbitrary von Neumann algebras. We give a 2-{$C^*$}-categorical/planar algebraic description of normal faithful conditional expectations $E:\mathcal{M}\to\mathcal{N}\subset\mathcal{M}$ with finite index and their duals $E':\mathcal{N}'\to\mathcal{M}'\subset\mathcal{N}'$ by means of the solutions of the conjugate equations for the inclusion morphism $\iota:\mathcal{N}\to\mathcal{M}$ and its conjugate morphism $\overline{\iota}:\mathcal{M}\to\mathcal{N}$. In particular, the theory of index for conditional expectations admits a 2-{$C^*$}-categorical formulation in full generality. Moreover, we show that a pair $(\mathcal{N}\subset\mathcal{M}, E)$ as above can be described by a Q-system, and vice versa. These results are due to Longo in the subfactor/simple tensor unit case [Lon90, Thm.\ 5.2], [Lon94, Thm.\ 5.1].