I am reading Parallel transport on principal bundles over stacks. I quote from their paper :

Let $G$ be a Lie group and $M$ a $C^{\infty}$ manifold. Recall that a choice of a connection $1$-form $A\in \Omega^1(P,\mathfrak{g})^G$ on a principal $G$-bundle $P$ over the manifold $M$ and a choice of base point $x\in M$ gives rise to the holonomy map $$\Omega(M,x)\rightarrow \text{Aut (fiber of $P$ at $x$)}\cong G,$$ where $\Omega(M,x)$ is the set of smooth loops at $x$ in $M$.

I do not have a probelm with above set up, except that I do not know how they are identifying $\text{Aut (fiber of $P$ at $x$)}\cong G$. I think along with fixing a point $x\in M$, they are also fixing a point $u\in \pi^{-1}(x)$, then, I know what is the map $\Omega(M,x)\rightarrow G$. If some one can clarify about it, it is good. But that is not the question. Question is from another line in the paper :

For a connected manifold $M$ holonomy map uniquely determines the connection $A$, and infact the bundle $P$ itself.

The reference they gave is

Shoshichi Kobayashi. La connexion des varietes fibrees. I, II. C. R. Acad. Sci. Paris, 238: 318–319, 443–444, 1954.

It is in a language that I can not read.

Can some one give an English reference where this is proved or a sketch of the proof is given or can some one write a sketch of the proof here?

Edit : As pointed in comments, there is an obvious way to identify $\text{Aut (fiber of $P$ at $x$)}\cong G$ if $G$ is abelian. In case $G$ is non abelian, can some one point me to a reference where this identification is explained.

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