Categorical Approach to Orbit–Stabilizer Theorem

Let $X$$X$ be a $G\text{-}\mathsf{Set}$$G\text{-}\mathsf{Set}$. We could view it as a groupoid by take $\mathrm{Ob}(\hat{X})=X,\mathrm{Mor}(\hat{X})=G$$\mathrm{Ob}(\widehat X)=X,\mathrm{Mor}(\widehat X)=G$. i.e. $x⟼g\cdot x$$x\longmapsto g\cdot x$ as morphism.

Then we have

$$\coprod _{y\in X}\mathrm{Hom}(x,y)=\coprod _{y\in G\cdot x}\mathrm{Hom}(x,y)$$

Now we have

$$\mathrm{\forall }y\in G\cdot x,x\cong y⟹\mathrm{Hom}(x,y)\cong \mathrm{Hom}(x,x)=G_x$$

Thus

$$\coprod _{y\in G\cdot x}\mathrm{Hom}(x,y)\cong \coprod _{y\in G\cdot x}\mathrm{Hom}(x,x)$$

The LHS is $G$$G$ and the RHS is $\begin{array}{c}\coprod _{y\in G\cdot x}{G}_x\end{array}$$\coprod_{y\in G\cdot x}G_x\\$.

Hence we have

$$|G|=|G\cdot x||G_x|$$