We provide sufficient conditions for mathematically rigorous proofs of the
third order universal laws capturing the energy flux to large scales and
enstrophy flux to small scales for statistically stationary, forced-dissipated
2d Navier-Stokes equations in the large-box limit. These laws should be
regarded as 2d turbulence analogues of the $4/5$ law in 3d turbulence,
predicting a constant flux of energy and enstrophy (respectively) through the
two inertial ranges in the dual cascade of 2d turbulence. Conditions implying
only one of the two cascades are also obtained, as well as compactness criteria
which show that the provided sufficient conditions are not far from being
necessary. The specific goal of the work is to provide the weakest
characterizations of the '0-th laws' of 2d turbulence in order to make
mathematically rigorous predictions consistent with experimental evidence.