I've come across two definitions of an equivalence of étale Lie groupoids, and I'd like to know whether they are equivalent.


Let $\mathcal{G}$ be an étale Lie groupoid with space of objects $\mathcal{G}\_0$ and space of arrows $\mathcal{G}\_1$.
Write $\alpha$ and $\omega\colon \mathcal{G}\_1 \to \mathcal{G}\_0$ for the maps sending an arrow
to its source and target, respectively.


**Definition 1.**
In *Metric Spaces of Nonpositive Curvature,* Bridson–Haefliger define an *equivalence*
of étale (Lie) groupoids $f\colon \mathcal{G} \to \mathcal{H}$ to be a smooth functor such that
$f\colon \mathcal{G}\_0 \to \mathcal{H}\_0$ is an étale map (a local diffeomorphism) and such that


1. for all $x \in \mathcal{G}\_0$, the map $f$ induces an isomorphism of isotropy groups
$\mathcal{G}\_x \to \mathcal{H}\_{f(x)}$, and
2. the map $f$ induces a bijection of orbit spaces $\mathcal{G}\_1\backslash\mathcal{G}\_0 \to \mathcal{H}\_1\backslash\mathcal{H}\_0$.


**Definition 2.** In ["Orbifolds as Groupoids: An Introduction"](https://arxiv.org/pdf/math/0203100.pdf),
Moerdijk defines an equivalence of (étale) Lie groupoids $f\colon\mathcal{G} \to \mathcal{H}$
to be a smooth functor such that


1. the map $\omega\pi\_1 \colon \mathcal{H}\_1\times\_{\mathcal{H}\_0}\mathcal{G}\_0 \to \mathcal{H}\_0$ defined on the fiber product $\{(g,y) \in \mathcal{H}\_1\times\mathcal{G}\_0 : \alpha(g) = f(y)\}$ sending a pair $(g,y)$ to $\omega(g)$ is a surjective submersion, and
2. the following diagram is a pullback square $\require{AMScd}$
$$\begin{CD} \mathcal{G}\_1 @>f>> \mathcal{H}\_1 \\
@VV{\alpha\times\omega}V @VV{\alpha\times\omega}V \\
\mathcal{G}\_0\times\mathcal{G}\_0 @>{f\times f}>> \mathcal{H}\_0\times\mathcal{H}\_0,
\end{CD}$$
i.e. the map $g \mapsto (\alpha(g),\omega(g),f(g))$ is a diffeomorphism from $\mathcal{G}\_1$
to $\{ (x,y,g) \in \mathcal{G}\_0\times\mathcal{G}\_0\times\mathcal{H}\_1 : \alpha(g) = f(x),\
\omega(g) = f(y)\}$.


**Remark.**
Moerdijk notes that the first condition says that each $x \in \mathcal{H}\_0$ is isomorphic to some $f(y)$, so the map of orbit spaces is a surjection. He also notes that the second condition implies $f$ induces a diffeomorphism from the space of arrows $g \colon x \to y$ in $\mathcal{G}$ to the space of arrows $g' \colon f(x) \to f(y)$ in $\mathcal{H}$, so the map of orbit spaces is an injection, and for each $x \in \mathcal{G}\_0$, the map $f$ induces an isomorphism of isotropy groups $\mathcal{G}\_x \to \mathcal{H}\_{f(x)}$. Thus Definition 2 implies all parts of Definition 1 except possibly the requirement that $f$ is an étale map.



Equivalence of definitions of equivalence of étale Lie groupoids
Notice that because $\omega$ is étale and $\omega\pi\_1$ is a submersion, $\pi\_1\colon \mathcal{H}\_1\times\_{\mathcal{H}\_0}\mathcal{G}\_0 \to \mathcal{H}\_1$ is a submersion, as is $\alpha\pi\_1$. Since $\alpha$ is étale, $\pi\_2\colon \mathcal{H}\_1\times\_{\mathcal{H}\_0}\mathcal{G}\_0 \to \mathcal{G}\_0$ is étale. Therefore because the composition $\alpha\pi\_1 = \pi\_2f$ is a submersion, so is $f \colon \mathcal{G}\_0 \to \mathcal{H}\_0$. Because $f$ induces a bijection of orbit spaces, only points in the same orbit may be identified by $f$. Since $\mathcal{G}$ is étale, its orbits are discrete. Therefore the dimension of $\mathcal{G}\_0$ is equal to the dimension of $\mathcal{H}\_0$ and we conclude that $f\colon \mathcal{G}\_0 \to \mathcal{H}\_0$ is étale. (Actually, $f\colon \mathcal{G}\_1 \to \mathcal{H}\_1$ is also étale). Therefore Definition 2 implies Definition 1.


**Edit: Definition 1 implies Definition 2.** Consider the fiber product
$$\begin{CD} \mathcal{H}\_1\times\_{\mathcal{H}\_0}\mathcal{G}\_0
@>{\pi\_2}>> \mathcal{G}\_0 \\
@V{\pi\_1}VV @VfVV \\
\mathcal{H}\_1 @>\alpha>> \mathcal{H}\_0.
\end{CD}$$
Because $f$ is étale, $\pi\_1$ is étale; therefore $\omega\pi\_1$ is *a fortiori* a submersion. It is surjective as soon as, for every $x \in \mathcal{H}\_0$, there exists $y\in \mathcal{G}\_0$ and an arrow $f(y) \to x$. This is true because $f$ induces a bijection of orbit sets.


Now consider the fiber product
$$\begin{CD}
K = (\mathcal{G}\_0\times\mathcal{G}\_0)\times\_{\mathcal{H}\_0\times\mathcal{H}\_0}\mathcal{H}\_1 @>{\pi\_2}>> \mathcal{H}\_1 \\
@V{\pi\_1}VV @V{\alpha\times\omega}VV \\
\mathcal{G}\_0 \times\mathcal{G}\_0 @>{f\times f}>> \mathcal{H}\_0\times\mathcal{H}\_0.
\end{CD}$$


We want to show that the map $F\colon \mathcal{G}\_1 \to K$ defined by
$g \mapsto (\alpha(g),\omega(g),f(g))$ is a diffeomorphism. Notice that because $f\times f$ is étale and $\pi\_2F = f$ is étale, $F$ is étale.
We need therefore only show that it is a bijection. It is easy to argue that because $f$ induces a bijection of orbit sets and isomorphisms on isotropy groups, for each $x$ and $y \in \mathcal{G}\_0$, the map $f$ induces a bijection from the set $\{g\colon x \to y\}$ to the set $\{g' \colon f(x) \to f(y)\}$. Therefore $F$ is a diffeomorphism.


