Triples of infinite iterates of metrizable functors

The concepts of metrizable, uniformly metrizable, and perfectly metrizable functors are introduced. The triple $\mathscr F^+$, $\mathscr F^{++}$, $\mathscr F^\omega$ of infinite iterates of a perfectly metrizable functor $\mathscr F$ is defined. The geometric properties of such triples are investigated for the continuous hyperspace functor and the probability measure functor.