  We analyze the axial $\gamma W$-box diagram for $I(J^P)=1(0^+)$ nuclei and
provide a dispersion representation of the nuclear-structure correction
$\delta_\text{NS}$ including its energy-dependent part. We also summarize
useful isospin rotation formula and representations in nuclear theory that
could facilitate the calculation of the parity-odd nuclear structure function
$F_3(\nu,Q^2)$. They provide a rigorous theory framework for the future,
high-precision calculation of the nuclear structure correction
$\delta_\text{NS}$ necessary for the extraction of the
Cabibbo-Kobayashi-Maskawa matrix element $|V_{ud}|$ from superallowed nuclear
$\beta$ decays.
