I am reading "Some notes on Differentiable stacks" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows.

Let $G$ be a Lie group action on a manifold $X$ (left action). We define the quotient stack $[X/G]$ as $[X/G](Y):=\{P\xrightarrow{p} Y, P\xrightarrow{f}X | P\rightarrow Y \text{ is a G-bundle,} ~ f \text{ is } G\text{-equivariant}\}$.

Morphisms of objects are $G$-equivariant isomorphisms.

I am trying to understand the motivation for defining in this way.

Given a Lie group action of $G$ on $X$, if I want to associate a stack, I would start with simpler cases which allows me to guess how to define.

Given a Lie group $G$, I have stack associated to it, denoted by $BG$, the stack of principal $G$ bundles.

Given a manifold $M$, I have the stack associated to it, denoted by $\underline{M}$ whose objects are maps $Y\rightarrow M$.

Suppose $X$ is trivial and $G$ acts trivially on $X=\{*\}$ then $[X/G]$ should only depend on $G$. We know what stack to associate for a Lie group $G$ i.e., $BG$. Thus, $[X/G]$ should just be $BG$.

Suppose $G$ is trivial and $G$ acts on $X$, $[X/G]$ should only depend on $X$. We know what stack to associate for a manifold $X$ i.e., $\underline{X}$. Thus, $[X/G]$ should just be $\underline{X}$.

Suppose $G$ is non trivial and $X$ is non trivial and that the action of $G$ on $X$ is such that $X/G$ is a manifold. We know what stack to associate for a manifold $X/G$ i.e., $\underline{X/G}$. Thus, $[X/G]$ should just be $\underline{X/G}$.

I am not able to guess how could we guess the definition knowing above three cases. Is quotient stack definition motivated from these simpler cases or Is it the case that simpler cases are special cases of notion of Quotient stack. How could we come up with such a definition. Any comments regarding the motivation are welcome.

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