Let $\mathcal A$ be a completely rational local Möbius covariant net on $S^1$, which describes a set of chiral observables. We show that local Möbius covariant nets $\mathcal B_2$ on 2D Minkowski space which contain the chiral theory $\mathcal A$ are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category $\mathrm{DHR}(\mathcal A)$. The Möbius 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.