I am reading Orbifolds as Groupoids: an Introduction by Ieke Moerdijk.

In page $8$ when explaining local charts, it says the following :

Let $\mathcal{G}$ be a Lie groupoid. For an open set $U\subseteq \mathcal{G}_0$, we write $\mathcal{G}|_U$ for the full subgroupoid of G with $U$ as a space of objects. In other words, $(\mathcal{G}|_U)_0=U$ and $(\mathcal{G}|_U)_1=\{x\rightarrow y : x,y\in U\}$. If $\mathcal{G}$ is proper and etale, then for each $x\in \mathcal{G}_0$ there exist arbitrary small neighborhoods $U$ of $x$ for which $\mathcal{G}|_U$ is isomorphic to $\mathcal{G}_x\ltimes U$ for an action of the isotropy group $\mathcal{G}_x$ on $U$.

Reference given is Orbifolds, Sheaves and Groupoids by Ieke Moerdijk and D. A. Pronk.

In page $12$, discussing Characterization of Orbifolds, they gave a proof of this which I am not able to understand completely.

I will write down the proof that I understood.

Let $x\in \mathcal{G}_0$. We are expected to find an open set containing $x$ on which $\mathcal{G}_x$ acts.

Let $\gamma\in \mathcal{G}_x$. As $s,t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ are local diffeomorphisms, we can choose a common open set $W_\gamma$ of $\gamma$ such that $s|_{W_\gamma}:W_\gamma\rightarrow s(W_\gamma)$ and $t|_{W\gamma}:W_\gamma\rightarrow t(W_\gamma)$ are diffeomorphisms.

Consider the composition $s(W_\gamma)\xrightarrow{s^{-1}}W_\gamma\xrightarrow{t} t(W_\gamma)$. As $\gamma\in \mathcal{G}_x$, we have $x\in s(W_\gamma)\bigcap t(W_\gamma)$. So, given $\gamma\in \mathcal{G}_x$, we have open sets $s(W_\gamma),t(W_\gamma)$ containing $x$ and a diffeo. $s(W_\gamma)\rightarrow t(W_\gamma)$. Here, we have two possibly different open sets containing $x$ and these open sets as well as the map $s(W_\gamma)\rightarrow t(W_\gamma)$ vary with $\gamma$. What we want is a single open set $V$ containing $x$ for each $\gamma$ giving the action $\mathcal{G}_x\times V\rightarrow V$.

As first step, we can consider $U_x$ to be $\bigcap s(W_\gamma)$. We have for each $\gamma\in \mathcal{G}_x$ map $s(W_\gamma)\rightarrow t(W\gamma)$. So, if we take intersection, we get one single open set, that works (not completely in the sense we want) and we have $\mathcal{G}_x\times U_x\rightarrow \bigcup t(W_\gamma)$. There is no obvious why $U_x=\bigcup t(W_\gamma)$. So, we consider a smaller subspace $V_x$ of $U_x$ containing $x$ so that we get $\mathcal{G}_x\times V_x\rightarrow V_x$.

Choose $V_x\subset U_x$ such that $$(V_x\times V_x)\bigcap (s,t)\left( \mathcal{G}_1\setminus \bigcup W_\gamma\right)=\emptyset.$$

This is still not a good choice of open sets that gives action of $\mathcal{G}_x$.

Denote by $\tilde{\gamma}$ the map $s(W_\gamma)\rightarrow t(W_\gamma)$ defined above. As $V_x\subseteq s(W_\gamma)$, this gives a map $\tilde{\gamma}:V_x\rightarrow t(W_\gamma)$.

We want to produce an open subset $N_x\subseteq V_x$ containing $x$ such that $\mathcal{G}_x$ acts on $N_x$ i.e., for each $\delta\in \mathcal{G}_x$ and $y\in N_x$ we should have $\delta.y=\tilde{\delta}(y)\in N_x$. This suggests a choice for $N_x$, namely $$N_x=\{y\in V_x: \tilde{\delta}(y)\in V_x \forall \delta\in \mathcal{G}_x\}.$$ We see that $\mathcal{G}_x$ acts on $N_x$. Let $\gamma\in \mathcal{G}_x$ and $y\in N_x$. We see that $\tilde{\gamma}(y)\in N_x$ i.e., $\tilde{\delta}(\tilde{\gamma}(y))\in V_x$ for each $\delta\in \mathcal{G}_x$.

As $y\in N_x$, we have $\tilde{\delta'}(y)\in V_x$ for each $\delta'\in\mathcal{G}_x$, in particular $\tilde{\delta\gamma}(y)=\tilde{\delta}(\tilde{\gamma}(y))\in V_x$ for each $\delta\in \mathcal{G}_x$. Thus, $\tilde{\gamma}(y)\in N_x$. Thus, we have an action of $\mathcal{G}_x$ on $N_x$.

It then says,

The restriction of the groupoid $\mathcal{G}|_{N_x}$ is exactly the translation groupoid of the action of $\mathcal{G}_x$ on $N_x$.

I am not able to see how does that follow. Any hints would be useful.

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