We have that $G-PBun(X)$, the category of topological principal bundles for a structure group $G$ is equivalent to $Top[X,BG]$ where $BG$ is the classifying space of $G$.
This almost looks like an adjunction - can it be turned into one?
Maybe in a higher-dimensional sense?
On
There is no left adjoint of $B$ as far as I know, which seems like what youre asking for, but there is in fact a right adjoint. Let $\Omega$ denote the loop space functor sending a space $S$ to the space of based loops in $S$. Then $\Omega$ is a functor to topological monoids where based loops are composed in the obvious way and we have a natural isomorphism
$$ \operatorname{Hom}(BG, X) \cong \operatorname{Hom}(G,\Omega X) $$
where the left $\operatorname{Hom}$ is as pointed topological spaces and the right $\operatorname{Hom}$ is as topological monoids.
This no longer classifies principal $G$-bundles since were looking at maps out of $BG$ but it is an adjunction of the $B$ functor.
For a reference, see this question.
This is not an adjunction, but rather the statement that the $2$-functor which sends a (nice) space $X$ to its groupoid of principal $G$-bundles is representable, namely by $BG$. Actually this is the functorial definition of $BG$.