  Despite the robustness of the chiral edge modes of quantum Hall systems
against the superconducting proximity effect, Cooper pairs can penetrate into
the chiral edge channels and carry the Josephson current in an appropriate
setup. In our work, the Josephson junction of a spin-polarized quantum
anomalous Hall insulator (QAHI) with a Chern number $\nu=1$ connecting
conventional superconductors is studied from the perspective of pairing
symmetry consistent with the chiral edge mode. Induced pairing states are
equal-spin triplet, a combination of the even- and odd-frequency components,
nonlocally extended, and have a finite momentum $2k_F$. The signature of the
equal-spin triplet pairings is confirmed via the dependence on the
interface-magnetization direction, and that of the finite-momentum pairing
states via the spatial profile of the anomalous Green's function. In the
presence of disorder, the robustness of the chiral edge mode leads to high
sensitivity of the critical current and the equilibrium phase difference to
disorder configurations, which is resulting from the interference of
current-carrying channels. The numerical calculations on a lattice model are
also examined by a simplified analytical model.
