  Let $\mathbf{X}$ be a class of metric spaces and let
$\mathbf{P}_{\mathbf{X}}$ be the set of all $f:[0, \infty)\to [0, \infty)$
preserving $\mathbf{X},$ $(Y, f\circ\rho)\in\mathbf{X}$ whenever $(Y,
\rho)\in\mathbf{X}.$ For arbitrary subset $\mathbf{A}$ of the set of all metric
preserving functions we show that the equality
$\mathbf{P}_{\mathbf{X}}=\mathbf{A}$ has a solution iff $\mathbf{A}$ is a
monoid with respect to the operation of function composition. In particular,
for the set $\mathbf{SI}$ of all amenable subadditive increasing functions
there is a class $\mathbf{X}$ of metric spaces such that
$\mathbf{P}_{\mathbf{X}}=\mathbf{SI}$ holds, which gives a positive answer to
the question of paper [1].
