Differentiable stacks and morita morphism I heard that if $[X_0/G_0]$ and $[X_1/G_1]$ are differentiable stacks, then any morphism between them is naturally equivalent to 
$$(G_0 \rightrightarrows X_0) \xleftarrow{\simeq} (G_2 \rightrightarrows X_2) \xrightarrow{F} (G_1 \rightrightarrows X_1)$$
where the left arrow is morita morphism and the right one is a Lie groupoid morphism.
I can't find a proof of this. Where can I find a detailed proof of this proposition?
 A: Given a Lie groupoid $\mathcal{G}=(\mathcal{G}_1\rightrightarrows \mathcal{G}_0)$ we have what is called $B\mathcal{G}$ the stack associated to $\mathcal{G}$ which is collection of principal $\mathcal{G}$ bundles. This in your notation is $[\mathcal{G}_0/\mathcal{G}_1]$.
Let $F:B\mathcal{G}\rightarrow B\mathcal{H}$ be a morphism of stacks. This stack $B\mathcal{G}$  has a special object $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$, the target map.
Consider image of $t$ under $F$, you get a principal $\mathcal{H}$ bundle $F(t:\mathcal{G}_1\rightarrow \mathcal{G}_0)$ of the form $P\rightarrow \mathcal{G}_0$ (as $F$ is fibre preserving, base manifold of $t$ and that of $F(t)$ has to be same).
As $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ has a left action of $\mathcal{G}$ and as $F$ is a functor, $F(t):P\rightarrow \mathcal{G}_0$  would also have a left action.
This gives a $\mathcal{G}-\mathcal{H}$ bibundle $$(\mathcal{G}_1\rightrightarrows \mathcal{G}_0)\leftarrow P\rightarrow (\mathcal{H}_1\rightrightarrows \mathcal{H}_0)$$
This bibundle is also called as generalzied morphism and this comes from 
$$(\mathcal{G}_1\rightrightarrows \mathcal{G}_0)\leftarrow (P\rightrightarrows M )\rightarrow (\mathcal{H}_1\rightrightarrows \mathcal{H}_0)$$
You can look at Orbifolds as Stacks by Eugene Lerman. Do not think this notes is about orbifolds. The word orbifold occurs only once in the note and that too in last  but one line of the paper. 
