Explicit description of classifying space as a "delooping"?

Question

Consider the "classifying-space functor" $\mathscr{B} : \mathrm{TopGrp} \rightarrow \mathrm{Top}$, constructed in the standard way as a geometric realization of a nerve (in TopCat if necessary) of the topological one-object groupoid corresponding to the inputted topological group $G$.

If I'm not mistaken, we may assume we are restricting $\mathrm{Top}$ to some "nice" category of nice topological spaces, closed under e.g. loop spaces, taking classifying spaces of topological groups, etc.

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The nLab page on delooping strongly suggests that $\mathscr{B}$ produces a "delooping" of (the underlying topological spaces of) topological groups in the context of $\mathrm{Top}$ as an $(\infty,1)$-topos with homotopy pullbacks.

I was wondering, is there an explicit description of a weak(?) homotopy equivalence $$ G \xrightarrow{\sim} \Omega(\mathscr{B}G) $$ natural (up to "higher homotopy"?) in the topological group $G$?

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I can't seem to find an explicit description of the above natural homotopy equivalence, although it's clear why the homotopy groups agree on both sides.

So far, I've tried to apply classifying space theory: since $[G,\Omega \mathscr{B} G] \simeq [\Sigma G , \mathscr{B}G]$, we are therefore looking for some particular topological principal $G$-bundle on the reduced suspension $\Sigma G$ (up to isomorphism), however I'm not sure how to find such a bundle.

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Update: Actually the answer comes from Hatcher's Algebraic Topology, Proposition 4.66; see the answer below.

Answer 1

Sorry, I realized the answer quickly follows from Hatcher's Algebraic Topology, Proposition 4.66

If $F \rightarrow E \rightarrow B$ is a fibration or fiber bundle with $E$ contractible, then there is a weak homotopy equivalence $F \rightarrow \Omega B$.

applied to the universal bundle $G \rightarrow\mathscr{E}G\rightarrow \mathscr{B}G$.