Given a finite group $G$, and a finite category $\mathcal{C}$, one can define the action of $G$ on $\mathcal{C}$ as a functor $A_{\mathcal{C}}\colon G\to\mathbf{Cat}$, which takes the single object of $G$ (regarded as a category) to $\mathcal{C}$. Moreover, one can define the quotient $\mathcal{C}/G$ to be the colimit of $A_{\mathcal{C}}$. There is an explicit construction of the category $\mathcal{C}/G$: let's denote the orbit of an element $a$ by $Ga$. The object set of $\mathcal{C}/G$ is simply given by the orbits of the elements of $\mathcal{C}^{(0)}$. To construct the set of morphism ${\mathcal{C}/G}^{(1)}$, one defines a relation $\leftrightarrow$ on $\mathcal{C}^{(1)}$ by saying $f\leftrightarrow g$ iff there is are decompositions $f = f_1\circ\...\circ f_n$ and $g=g_1\circ\...\circ g_n$, such that $G f_i = G g_i$ for all $i=1,...,n$. This relation is clearly symmetric and reflexive. It is however, not transitive. So one defines $\sim$ to be the transitive closure of $\leftrightarrow$ and sets $\mathcal{C}/G^{(1)} := \mathcal{C}^{(1)}/\sim$. My problem is that even though I can imagine why transitivity fails (i.e. given 3 morphisms $f,g,h$, one might find decompostions such that $f\sim g$ and $g\sim h$, but no decompositions such that $f\sim h$), I can't find an explicit example to demonstrate that case.

P.S.: As far as I know this might very well work too without the restrictions to finite groups and categories. But I haven't thought this through yet. So I restricted my question to the finite case.