Lifting the functors $U_\tau$ and $U_R$ to the categories of bounded metric spaces and uniform spaces

Metric and uniform properties of the unit ball functors $U_\beta$, $U_R$, $U_\tau$ of measures with compact support, Radon measures, and $\tau$-additive measures, respectively, are studied. It is proved that these functors can be lifted to the category $\mathbf{BMetr}$ of bounded metric spaces, $\mathbf{BMetr}_u$ of bounded metric spaces and uniformly continuous maps, and $\mathbf{Unif}$ of uniform spaces. Additionally, it is shown that the functor $U_\tau$ preserves the completeness property of metric spaces.