Delhomme-Laflamme-Pouzet-Sauer space as groupoid

Let ${\mathbb{R}}^{+}=[0,\mathrm{\infty })$ and let $d^{+}$ be the ultrametric on ${\mathbb{R}}^{+}$ such that $d^{+}(x,y)=\max \{x,y\}$ for all different $x,y\in {\mathbb{R}}^{+}$. It is shown that the monomorphisms of the groupoid $({\mathbb{R}}^{+},d^{+})$ coincide with the injective ultrametric-preserving functions and that the automorphisms of $({\mathbb{R}}^{+},d^{+})$ coincide with the self-homeomorphisms of ${\mathbb{R}}^{+}$. The structure of endomorphisms of $({\mathbb{R}}^{+},d^{+})$ is also described.