Tale of two- U ( 1 ) axion models

$U(1)$ global symmetry to solve the strong $CP$ problem could be a remnant of multi-$U(1)$ symmetries from QCD and hidden strong dynamics. Both Peccei-Quinn $U(1)$ and dynamical $U(1)$ are described uniformly, based on which we classify various mixed two-$U(1)$ models to solve both strong $CP$ and quality problems. We propose a moose diagram method with different fermion assignments to directly read relations between $CP$ phases, which illustrate how the strong $CP$ problem is solved in terms of cancellation between $CP$ phases. In two-axion models, we find that the lightest axion is still the same as a QCD axion in the infrared region, while a one-axion model with ${Z}_{2}$ symmetry enhances the axion mass spectrum. Our discussions can be extended to multiaxion cases.