On projectively quotient functors

We introduce notions of projectively quotient, open, and closed functors. We give sufficient conditions for a functor to be projectively quotient. In particular, any finitary normal functor is projectively quotient. We prove that the sufficient conditions obtained are necessary for an arbitrary subfunctor $\mathcal{F}$ of the functor $\mathcal{P}$ of probability measures. At the same time, any “good” functor is neither projectively open nor projectively closed.