On monoids of metric preserving functions

Let X be a class of metric spaces and let PX be the set of all f:[0, ∞)→ [0, ∞) preserving X, (Y, f∘ρ) whenever (Y, ρ). For arbitrary subset A of the set of all metric preserving functions we show that the equality PX=A has a solution iff A is a monoid with respect to the operation of function composition. In particular, for the set SI of all amenable subadditive increasing functions there is a class X of metric spaces such that PX=SI holds, which gives a positive answer to the question of paper [1].