Ultrametric Preserving Functions and Weak Similarities of Ultrametric Spaces$$^*$$

Let $WS(X, d)$ be the class of ultrametric spaces which are weakly similar to space $(X, d)$. The main results paper completely describe for equality $$ \rho(x, y) = f(d(\Phi(x), \Phi(y))) holds every $(Y, \rho) \in WS(X, d)$, weak similarity $\Phi \colon Y \to X$, and all $x$, $y Y$ with some (pseudoultrametric) preserving function $f$ depending on $\Phi$.