  We give a complete classification of isoparametric hypersurfaces in a product
space $M^2_{\kappa_1}\times M^2_{\kappa_2}$ of $2$-dimensional space forms for
$\kappa_i\in \{-1,0,1\}$ with $\kappa_1\neq \kappa_2$. In fact we prove that
any isoparametic hypersurface in such a space has constant product angle
function, which enables us to remove the condition of constant principal
curvatures from the classification obtained recently by J.B.M.dos Santos and
J.P.dos Santos.
