Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This Wikipedia article (or also this related MO question) defines a quotient stack $[X/G]$ as a category of principal $G$-bundles fibred over Sch/S.

When they discuss the map $[X/G] \to X/G$ (when $X/G$ exists as an algebraic space, or let's say scheme for convenience if one prefers), they say "complete the diagram" i.e. if $D\to T$ is a principal $G$-bundle corresponding to a $T$-point $T\to [X/G]$, then we can complete the diagram $T \leftarrow D\to X \to X/G$ (which I understand as induing map $T \to X/G$)

It is not clear to me how one can induce a map $T\to X/G$ using the fact that $D\to X$ is $G$-equivariant.

**Q1. How do we get the induced map $T\to X/G$?**

**Q2. If we have $T$-point of $X/G$, pull back of the diagram $T\to X/G \leftarrow X$ gives a principal $G$-bundle $D\to T$ and $D\to X$ which is $G$-equivariant. Then what is the obstruction of this map $X/G \to [X/G]$ being the quasi-inverse of $[X/G]\to X/G$?**