Definition
Suppose \(G\) is a group acting on a set \(X\). The moduli category of groupoids is the category \([X/G]\) whose objects are all elements \(x\in X\) and whose morphisms are \(\Hom(x,x') = \{g\in G ~|~ x' = gx\}\).
Remark: There is an equivalence of categories \([X/G] \simeq \mathcal C_{X/G}\) if and only if the action of \(G\) on \(X\) is free. See this exercise from Alper.