Let $G$ and $H$ be two *topological* groupoids. Recall that a morphism $F \colon H \to G$ is called an *essential equivalence*, if

- the map $t \circ \pi_1 \colon G_1 \times_{G_0} H_0 \to G_0$ is an open surjection, where $G_1 \times_{G_0} H_0$ is the pullback along the source map $s \colon G_1 \to G_0$.
- the following diagram is a pullback diagram: $$ H_1 \to G_1 \\\\ \downarrow \qquad \downarrow \\\\ H_0 \times H_0 \to G_0 \times G_0 $$ where the vertical arrow are given by $(s,t)$.

Basically this means that $F$ is an equivalence of categories (where the first condition implies essential surjectivity and the second it fullness), but it need not necessarily have a continuous inverse. Nevertheless, such a map $F$ induces a weak homotopy equivalence $BF \colon BH \to BG$.

I think, I know how to prove this. But for a paper I would rather have a reference in the literature for this result. Unfortunately, this seems to be so well-known that nobody bothers to write down a proof.

What is the best/first reference in the literature for the above statement?