The property $O_n$ for finite seminormal functors

We give a finite combinatorial test for finite seminormal functors to possess the property $O_n$ and use it in establishing that in some cases this property leads to some well-known functors. For example, if some functor $F$ possesses the property $O_n$ then $F_2$ coincides with either $\exp_2$ or the squaring functor. Hence we conclude that if $F(D^{\omega_1})$ фтв $D^{\omega_1}$ are homeomorphic then $F_2$ is either $\exp_2$ or $(\,\cdot\,)^2$.