  In [4], we proved that every noncompact ancient $\kappa$-solution to the
Ricci flow in dimension $3$ is either locally isometric to a family of
shrinking cylinders, or isometric to the Bryant soliton. In the same paper, we
announced that the same method implies that compact ancient $\kappa$-solutions
are rotationally symmetric. In this note we provide the details of this
argument.
