Consider as a representative of its orbit . Because the translation groupoid is equivalent to its skeleton, we must have \(\Hom_{\cat{sk}\cT_GX}(O_x,O_x)\cong\Hom_{\cT_GX}(x,x) =: G_x,\) the set of automorphisms of . This group consists of precisely those so that . In other words, the group $\Hom_{\cT_GX}(x,x)$ is the stabilizer of with respect to the -action. Note that this argument implies that any pair of elements in the same orbit must have isomorphic stabilizers. As is always the case for a skeletal groupoid, there are no morphisms between distinct objects. In summary, the skeleton of the translation groupoid, as a category, is the disjoint union of the stabilizer groups, indexed by the orbits of the action of on .