Let $\mathcal{G}$ be a Lie groupoid. The target map $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ is a principal $\mathcal{G}$ bundle. 

This article [Orbifolds as Stacks?][1] by Eugene Lerman calls (in page $11$) this particular principal bundle to be the unit principal $\mathcal{G}$ bundle. So, for a Lie groupoid $\mathcal{G}$, this $\mathcal{G}$ bundle is a special element in $B\mathcal{G}$.


Let $\mathcal{G}$ and $\mathcal{H}$ be Lie groupoids.
For $B\mathcal{G}$, I have a special element $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$. For $B\mathcal{H}$, I have a special element $t:\mathcal{H}_1\rightarrow \mathcal{H}_0$. 

Is it the case that any morphism of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$ should take this special element $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ to the special element $t:\mathcal{H}_1\rightarrow \mathcal{H}_0$? I could not see why this is true from definition of map of stacks but I feel this should be the case. Any comments are welcome.
 
Edit : A stack (over the category of manifolds $\text{Man}$) for me is a category $\mathcal{D}$ along with a functor $\mathcal{D}\rightarrow \text{Man}$ such that it is a category fibered in groupoids and some extra conditions. By an element of stack I mean an element (object) in the category $\mathcal{D}$. 
 

  [1]: https://arxiv.org/pdf/0806.4160.pdf