Update: I'm now at this step:
I want to confirm that if $b:P\to T$, then $[U/G](b)((P,f))$ is $P\times_TP$ and $P\times_TP\simeq P\times G$
Specifically, I want to understand in
where $\mathcal{P}$ is a $G$-torsor over $T$ and $f:\mathcal{P}\to U$ is $G$-equivariant (by Yoneda lemma), how is $T\times_{\mathcal{X}}U\simeq\mathcal{P}$ and $h=f$?