Example
Let \(G\) act on a set \(X\). Show that the moduli groupoid of orbits \([X/G]\) is equivalent (as a category) to the set \(X/G\) if and only if the action is free.
Proof
Assume the action is free and define a functor \(F:[X/G]\to \mathcal C_{X/G}\) by sending an object \(x\in X\) to the orbit \(\overline x \in X/G\) and a morphism \(g:x\to x'\) to the identity \(\overline x\to \overline x\). By freeness, \(\Hom_{[X/G]}(x,x')\) is either empty or a single element; so this functor is faithful. It is clearly full since \((\id :x\to x)\mapsto (\id : \overline x\to \overline x)\) and it is essentially surjective since it’s surjective on objects. Thus \(F\) is an equivalence of categories.
Now suppose we have an equivalence of categories \(F: [X/G]\to \mathcal C_{X/G}\). Since this is faithful, \(\Hom_{[X/G]}(x,x)\) must consist of exactly one element, the identity \(e\in G\). This implies the \(G\) action is free.