According to "[Notes on differentiable stacks](https://www.uni-due.de/~hm0002/stacks.pdf)" by Heinloth, 



> 
> the classifying stack will also classify $G$-bundles on stacks. (Remark 2.13)
> 
> 
> 


(Here $G$ is a Lie group.) My questions are:


(1) What is the precise statement? (The category of morphisms to the classifying stack $[pt/G]$ is equivalent to the category of principal $G$-bundles? If so, how do we define morphisms between principal $G$-bundles?)


(2) How do we prove the statement?


(3) Suppose we have a principal G-bundle over a differentiable stack $\mathcal{X}$. How do we construct a morphism from $\mathcal{X}$ to the classifying stack?


I would be most grateful if you could tell me any good references. 



Classification of principal G-bundles over a differentiable stack
They key insight is that the bicategory of differentiable stacks is equivalent to the bicategory of Lie groupoids, where the 1-morphisms are so-called bibundles, or Hilsum-Skandalis morphisms. Under this equivalence, the statement is the following:


Let $\mathcal{X}$ be a Lie groupoid, and let $BG$ the Lie groupoid that represents the stack $[pt/G]$. Then there is an equivalence of categories
$$
Bun\_G(\mathcal{X}) = Hom\_{LieGrpd}(\mathcal{X},BG).
$$


In fact, this equivalence is now tautological, thanks to the translation into the Lie groupoid world. 


