  Let $\mathbb{R}^{+}=[0, \infty)$ and let $\mathbf{End}_{\mathbb{R}^+}$ be the
set of all endomorphisms of the monoid $(\mathbb{R}^+, \vee)$. The set
$\mathbf{End}_{\mathbb{R}^+}$ is a monoid with respect to the operation of the
function composition $g \circ f$. It is shown that $g : \mathbb{R}^+ \to
\mathbb{R}^+$ is pseudoultrametric-preserving iff $g \in
\mathbf{End}_{\mathbb{R}^+}$. In particular, a function $f : \mathbb{R}^+ \to
\mathbb{R}^+$ is ultrametrics-preserving iff it is an endomorphism of
$(\mathbb{R}^+,\vee)$ with kernel consisting only the zero point. We prove that
a given $\mathbf{A} \subseteq \mathbf{End}_{\mathbb{R}^+}$ is a submonoid of
$(\mathbf{End}, \circ)$ iff there is a class $\mathbf{X}$ of pseudoultrametric
spaces such that $\mathbf{A}$ coincides with the set of all functions which
preserve the spaces from $\mathbf{X}$. An explicit construction of such
$\mathbf{X}$ is given.
