Definition
(Moduli groupoid of orbits). Given a group \(G\) acting on a set \(X\), the moduli groupoid of orbits \([X/G]\) is defined by taking the objects to be all elements \(x\in X\) and declaring \(\Hom(x, x') = \{g\in G \mid x' = gx\}\).
It is equivalent to a set (a groupoid in which only identity morphisms exist) if and only if the action of \(G\) on \(X\) is free, in which case it is equivalent to the set of orbits \(X/G\) viewed as the groupoid of a set.