Let $G_1\substack{\to \\ \to}G_0$ be an étale Lie groupoid, whose source- and target-maps are denoted by $s$ and $t$, respectively. Let \begin{equation} \Psi=\{(t|_U)\circ(s|_U)^{-1}:\text{$U$ is an open set of $G_1$, and $s$ and $t$ are injective on $U$}\}\text. \end{equation} A textbook I am reading (p.138 of "Introduction to foliations and Lie groupoids") claims that $\Psi$ is a pseudogroup on $G_0$. I checked everything except for the gluing-condition: For all diffeomorphism $f:V\to W$ between open sets of $G_0$, if there exists an open cover $(V_i)_{i\in I}$ of $V$ such that $f|_{V_i}\in\Psi$ for all $i\in I$, then $f\in\Psi$. How do you prove this?

Let $G_1\substack{\to \\ \to}G_0$ be an étale Lie groupoid, whose source- and target-maps are denoted by $s$ and $t$, respectively. Let \begin{equation} \Psi=\{(t|_U)\circ(s|_U)^{-1}:\text{$U$ is an open set of $G_1$, and $s$ and $t$ are injective on $U$}\}\text. \end{equation} A textbook I am reading (p.138 of "Introduction to foliations and Lie groupoids") claims that $\Psi$ is a pseudogroup on $G_0$. I checked everything except for the gluing-condition: For all diffeomorphism $f:V\to W$ between open sets of $G_0$, if there exists an open cover $(V_i)_{i\in I}$ of $V$ such that $f|_{V_i}\in\Psi$ for all $i\in I$, then $f\in\Psi$. I checked that the gluing-condition is true if $G_1\substack{\to \\ \to}G_0$ is effective. Is it still true without effectiveness?