  We study how the symmetry operator of the axial $U(1)$ non-invertible
symmetry acts on the 't~Hooft line operator in the $U(1)$ gauge theory by
employing the modified Villain-type lattice formulation. We model the axial
anomaly by a compact scalar boson, the ``QED axion''. For the gauge invariance,
the simple 't~Hooft line operator, which is defined by a line integral of the
dual $U(1)$ gauge potential, must be ``dressed'' by the scalar and $U(1)$ gauge
fields. A careful consideration on the basis of the anomalous Ward--Takahashi
identity containing the 't~Hooft operator with the dressing factor and a
precise definition of the symmetry operator on the lattice shows that the
symmetry operator leaves no effect when it sweeps out a 't~Hooft loop operator.
This result appears inequivalent with the phenomenon concluded in the continuum
theory. In an appendix, we demonstrate that the half-space gauging of the
magnetic $\mathbb{Z}_N$ 1-form symmetry, when formulated in an appropriate
lattice framework, leads to the same conclusion as above. A similar result is
obtained for the axion string operator.
