If $X$ is a topological space, $G$ a topological group and $E G \to BG$ a universal bundle, isomorphism classes of numerable principal $G$-bundles over $X$ are in one-to-one correspondence with homotopy classes of maps $X \to BG$.

I am curious if this fact can be refined to obtain a homotopic description of the category of principal $G$-bundles over $X$. In other words, is there some category of maps $X \to BG$ (or a variation of this) equivalent to the category of principal $G$-bundles over $X$?

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