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

- 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
- 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",
Moerdijk defines an equivalence of (étale) Lie groupoids $f\colon\mathcal{G} \to \mathcal{H}$
to be a smooth functor such that

- 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
- 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.