Naive question: Why $B:Mon\rightarrow Top^{*}, \Omega: Top^{*}\rightarrow Mon$ is an adjoint functor?

Question

It is not clear to me why we have a bijection of the form $$Mor_{Top^{*}}(BY,X)\rightarrow Mor_{Mon}(Y,\Omega X)$$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.

Answer 1

This question is answered in my post in mathoverflow.