On monoids of metric preserving functions

Let X be a class of metric spaces and let P_X be the set of all f : [0, ∞) → [0, ∞) preserving X, i.e., (Y, f ∘ ρ) ∈ X whenever (Y, ρ) ∈ X. For arbitrary subset A of the set of all metric preserving functions, we show that the equality P_X = A has a solution if 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 P_X = SI holds.