  Let $\mathcal{A}$ be a completely rational local M\"obius covariant net on
$S^1$, which describes a set of chiral observables. We show that local M\"obius
covariant nets $\mathcal{B}_2$ on 2D Minkowski space which contains
$\mathcal{A}$ as chiral left-right symmetry are in one-to-one correspondence
with Morita equivalence classes of Q-systems in the unitary modular tensor
category $\mathrm{DHR}(\mathcal{A})$. The M\"obius covariant boundary
conditions with symmetry $\mathcal{A}$ of such a net $\mathcal{B}_2$ are given
by the Q-systems in the Morita equivalence class or by simple objects in the
module category modulo automorphisms of the dual category. We generalize to
reducible boundary conditions.
  To establish this result we define the notion of Morita equivalence for
Q-systems (special symmetric $\ast$-Frobenius algebra objects) and
non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel,
namely that Rehren's construction (generalized Longo-Rehren construction,
$\alpha$-induction construction) coincides with the categorical full center.
This gives a new view and new results for the study of braided subfactors.
