  The COHERENT collaboration observed coherent elastic neutrino nucleus
scattering using a 14.6 kg cesium-iodide (CsI) detector in 2017 and recently
published the updated results before decommissioning the detector. Here, we
present the legacy determination of the weak mixing angle and of the average
neutron rms radius of $^{133}\mathrm{Cs}$ and $^{127}\mathrm{I}$ obtained with
the full CsI dataset, also exploiting the combination with the atomic parity
violation (APV) experimental result, that allows us to achieve a precision as
low as $\sim$4.5% and to disentangle the contributions of the
$^{133}\mathrm{Cs}$ and $^{127}\mathrm{I}$ nuclei. Interestingly, we show that
the COHERENT CsI data show a 6$\sigma$ evidence of the nuclear structure
suppression of the full coherence. Moreover, we derive a data-driven
APV+COHERENT measurement of the low-energy weak mixing angle with a percent
uncertainty, independent of the value of the average neutron rms radius of
$^{133}\mathrm{Cs}$ and $^{127}\mathrm{I}$, that is allowed to vary freely in
the fit. Additionally, we extensively discuss the impact of using two different
determinations of the theoretical parity non-conserving amplitude in the APV
fit. Our findings show that the particular choice can make a significant
difference, up to 6.5% on $R_n$(Cs) and 11% on the weak mixing angle. Finally,
in light of the recent announcement of a future deployment of a 10 kg and a
$\sim$700 kg cryogenic CsI detectors, we provide future prospects for these
measurements, comparing them with other competitive experiments that are
foreseen in the near future.
