$\DeclareMathOperator\At{At}\DeclareMathOperator\Obj{Obj}\DeclareMathOperator\Mor{Mor}$According to https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea the Atiyah Lie groupoid $\At(P)$ of a principal $G$ bundle $\pi:P \rightarrow X$ is a category such that $\Obj(\At(P))=\lbrace \pi^{-1}(x): x \in X \rbrace$ and $\Mor(\At(P))=\lbrace f:\pi^{-1}(x)\rightarrow \pi^{-1}(y): \text{$f$ is a $G$ equivariant morphism}\rbrace$. Structure maps of this category are easy to guess. Now it is easy to see that $\At(P)$ is indeed a groupoid.

Although it is mentioned in https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea that the Atiyah Lie groupoid is indeed a Lie groupoid, I am not able to guess appropriate smooth structures on $\Obj(\At(P))$ and $\Mor(\At(P))$ such that the source and the target maps are surjective submersions and other structure maps are smooth.

Is there any natural choice of such smooth structures on both $\Obj(\At(P))$ and $\Mor(\At(P))$ such that $At(P)$ is a Lie groupoid so that if someone talks about the Atiyah Lie groupoid of a principal $G$ bundle then he/she is precisely assuming those natural choice of smooth structures on $\Obj(\At(P))$ and $\Mor(\At(P))$?

I would also be very grateful if someone point me to any literature in this direction.

$\DeclareMathOperator\At{At}\DeclareMathOperator\Obj{Obj}\DeclareMathOperator\Mor{Mor}$According to https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea the Atiyah Lie groupoid $\At(P)$ of a principal $G$ bundle $\pi:P \rightarrow X$ is a category for which $$\Obj(\At(P))=\lbrace \pi^{-1}(x): x \in X \rbrace$$ and $$\Mor(\At(P))=\big\lbrace f:\pi^{-1}(x)\rightarrow \pi^{-1}(y): \text{$f$ is a $G$ equivariant morphism}\big\rbrace.$$ Structure maps of this category are easy to guess. Now it is easy to see that $\At(P)$ is indeed a groupoid.

Although it is mentioned in https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea that the Atiyah Lie groupoid is indeed a Lie groupoid, I am not able to guess appropriate smooth structures on $\Obj(\At(P))$ and $\Mor(\At(P))$ such that the source and the target maps are surjective submersions and other structure maps are smooth.

Is there any natural choice of such smooth structures on both $\Obj(\At(P))$ and $\Mor(\At(P))$ such that $At(P)$ is a Lie groupoid so that if someone talks about the Atiyah Lie groupoid of a principal $G$ bundle then he/she is precisely assuming those natural choice of smooth structures on $\Obj(\At(P))$ and $\Mor(\At(P))$?

I would also be very grateful if someone point me to any literature in this direction.

According to https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea Atiyah Lie Groupoid $At(P)$ of a Principal $G$ bundle $\pi:P \rightarrow X$ is a category such that $Obj(A(P))=\lbrace \pi^{-1}(x): x \in X \rbrace$ and $Mor(A(P))=\lbrace f:\pi^{-1}(x)\rightarrow \pi^{-1}(y): f$ is a $G$ equivariant morphism $\rbrace$. Structure maps of this category are easy to guess. Now it is easy to see that $At(P)$ is indeed a groupoid.

Although it is mentioned in https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea that Atiyah Lie Groupid is indeed a Lie Groupoid but I am not able to guess appropriate smooth structures on $Obj(At(P))$ and $Mor(At(P))$ such that the source and the target maps are surjective submersions and other structure maps are smooth.

Is there any natural choice of such Smooth structures on both $Obj(At(P))$ and $Mor(At(P))$ such that $At(P)$ is a Lie groupoid so that if someone talks about Atiyah Lie Groupoid of a Principal $G$ bundle then he/she is precisely assuming those natural choice of smooth structures on $Obj(At(P))$ and $Mor(At(P))$?

I would also be very grateful if someone point me to any literature in this direction.

Thank you.

$\DeclareMathOperator\At{At}\DeclareMathOperator\Obj{Obj}\DeclareMathOperator\Mor{Mor}$According to https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea the Atiyah Lie groupoid $\At(P)$ of a principal $G$ bundle $\pi:P \rightarrow X$ is a category such that $\Obj(\At(P))=\lbrace \pi^{-1}(x): x \in X \rbrace$ and $\Mor(\At(P))=\lbrace f:\pi^{-1}(x)\rightarrow \pi^{-1}(y): \text{$f$ is a $G$ equivariant morphism}\rbrace$. Structure maps of this category are easy to guess. Now it is easy to see that $\At(P)$ is indeed a groupoid.

Although it is mentioned in https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea that the Atiyah Lie groupoid is indeed a Lie groupoid, I am not able to guess appropriate smooth structures on $\Obj(\At(P))$ and $\Mor(\At(P))$ such that the source and the target maps are surjective submersions and other structure maps are smooth.

Is there any natural choice of such smooth structures on both $\Obj(\At(P))$ and $\Mor(\At(P))$ such that $At(P)$ is a Lie groupoid so that if someone talks about the Atiyah Lie groupoid of a principal $G$ bundle then he/she is precisely assuming those natural choice of smooth structures on $\Obj(\At(P))$ and $\Mor(\At(P))$?

I would also be very grateful if someone point me to any literature in this direction.