Classification of principal G-bundles by homotopy classes of maps into BG

Let $G$ be a Lie group and let $M$ be a paracompact smooth manifold. Write $\mathrm{Prin}_G(M)$ for the set of isomorphism classes of principal G-bundles over $M$ (isomorphisms are principal bundle isomorphisms covering $\mathrm{id}_M$ ).

Let $EG \to BG$ be the universal principal G-bundle over the classifying space $BG$ . For a continuous map $f:M\to BG$ , form the topological pullback $f^*EG=\{(x,e):f(x)=\pi(e)\}$ . It admits a smooth principal-bundle structure, unique up to smooth bundle isomorphism; this uses that $M$ is a smooth manifold and $G$ is a Lie group.

The assignment

[f] \in [M,BG] \longmapsto \big[f^*(EG)\big] \in \mathrm{Prin}_G(M)

is a well-defined bijection from the set of homotopy classes of maps $M\to BG$ to isomorphism classes of principal $G$ -bundles over $M$ .

Equivalently:

(Existence) For every principal $G$ -bundle $P\to M$ there exists a (continuous) classifying map $f_P\colon M\to BG$ such that $P \cong f_P^*(EG)$ as principal $G$ -bundles.
(Uniqueness) If $f,g\colon M\to BG$ are homotopic, then $f^*(EG)\cong g^*(EG)$ ; conversely, if $f^*(EG)\cong g^*(EG)$ then $f$ and $g$ are homotopic.

Examples

Contractible base. If $M$ is contractible, then $[M,BG]$ has one element, so every principal $G$ -bundle over $M$ is isomorphic to the trivial principal bundle.
Circle and disconnected structure group. Since unbased homotopy classes $[S^1,BG]$ correspond to conjugacy classes in $\pi_1(BG)\cong \pi_0(G)$ , principal $G$ -bundles over $S^1$ are classified by conjugacy classes in the component group $\pi_0(G)$ . In particular, they are all trivial when $G$ is connected. For instance, with $G=O(1)\cong \{\pm 1\}$ there are two classes; the nontrivial one corresponds (via the usual passage to associated line bundles) to the Möbius bundle.
Hopf bundle as a pullback. For $G=U(1)$ and $M=S^2$ , one has $[S^2,BU(1)]\cong \mathbb{Z}$ . Under this identification, the Hopf fibration represents a generator (so it is a pullback of $EU(1)\to BU(1)$ along a classifying map representing a generator).