We study some categorical properties of the functor Oβ of weakly additive functionalsacting in the category Tych of the Tychonoff spaces and their continuous mappings. We show that Oβ preserves the weight of infinite-dimensional Tychonoff spaces, the singleton, and the empty set, and that Oβ is monomorphic and continuous in the sense of T. Banakh, takes every perfect mapping to an epimorphism, and preserves intersections of functionally closed sets in a Tychonoff space.