Let $X$ be a paracompact Hausdorff space. An orbifold chart is a triple $(\tilde U, \phi, G)$, where $\tilde U\subseteq \mathbb{R^n}$ is open and connected, $G$ is a finite group acting by diffeomorphisms on $\tilde U$ and $(\tilde U, \phi, U)$ is a $G$-bundle, where $U\subseteq X$ is open and connected.
Let $(\tilde U_i,\phi_i,G_i)$ be two orbifold charts. They are locally compatible, if for every $x \in U_1\cap U_2$ there is some orbifold chart $(\tilde W, \psi, H)$ with $x\in W \subseteq U_1\cap U_2$ and embeddings $\lambda_i: (\tilde W,\psi,H) \to (\tilde U_i,\phi_i,G_i)$.
Is it true, that if $(\tilde V,\psi,G)$ is some orbifold chart and $(\tilde U_i, \phi_i,G_i) \to (\tilde V, \psi,G)$ are embeddings of charts, $i=1,2$, then $(\tilde U_1, \phi_1,G_1)$ and $(\tilde U_2, \phi_2,G_2)$ are locally compatible?
I wanted to show, that for orbifold atlases to have a common refinement is an equivalence relation. "Common refinement" is reflexive and symmetric, but transitive would require the above I think? Or am I completely wrong? I wanted to construct a common refinement $\mathcal{W}$ if I have two atlases $\mathcal{W_1}\rightarrow \mathcal{U} \leftarrow \mathcal{W_2}$, so that $\mathcal{W_1}\leftarrow\mathcal{W} \rightarrow \mathcal{W_2}$.