This is definitely not a research level question. I believe this is "common sense" among homotopists, however after "extensive" googling for 2 days I could not find a proof of it online or in standard textbooks (Hatcher, Milnor-Stasheff, Husemoller). I asked on mathstackexchange but did not receive an answer. I also tried to use online resources like nlab. It seems to be this is related to the bar construction, but I do not think the proof should be that complicated. I read in lecture notes that this is a natural consequence of the long exact sequence of fibration $F\rightarrow E\rightarrow B$, and the adjointness follows from homotopy lifting property, but I do not know how to put everything together. Below is the original post on Math.SE website. I also want to know "what should I read" for this kind of question, like why Bott periodicity can be framed as $\Omega U\cong \mathbb{Z}\times BU$.
It is not clear to me why we have a bijection of the form $$Mor_{Top^{*}}(BG,X)\rightarrow Mor_{Mon}(G,\Omega X)$$where $Mon$ is the category of topological monoids and $Top^{*}$ the based topological spaces. It seems to be something essentially trivial that the instructor did not bother to write down the proof in notes, but after thinking for 20minutes I still do not understand how to construct such a bijection to let $B$ and $\Omega$ be adjoint functors (or maybe I formulated it wrong somehow?). I think I need to this result to prove the well known result that $$\Omega BG\cong G, B\Omega X\cong X$$ Just to clarify definition, here $BG$ is the weakly contractible total space $EG$ quotient out by group action of $G$. $\Omega X$ is the group formed by mappings of the circle to $X$ with a fixed based point.
Sorry for the low level of this question.