Let Mon be the category of topological monoids. I am happy to work with the model structure mentioned here:

Model Structure/Homotopy Pushouts in topological monoids?.

I'm looking for a reference for the following statement, which I believe to be true. Suppose $X$ and $Y$ are topological monoids, with $X$ cofibrant, and $Y$ group-like.

**Assertion:** The map of spaces

$$ \text{monoid-maps}(X,Y) \to \text{maps}_\ast(BX,BY) $$

is a weak homotopy equivalence.

My guess is that one could prove this by induction, using the fact that $X$ is a retract of an object given by attaching free things. For example, here is a verification of the statement when $X = FU$, the free monoid on the points of a based space $U$. In this case,

$$ \text{monoid-maps}(FU,Y) = \text{maps}_\ast(U,Y) , $$

whereas $$ \text{maps}_*(BFU,BY) \simeq \text{maps}_\ast(\Sigma U,BY) $$ using, say James theorem $FU \simeq \Omega\Sigma U$. Since $Y$ is group-like, we have $Y \simeq \Omega BY$ and we get

$$ \text{maps}_*(\Sigma U,BY) \simeq \text{maps}_\ast(U,Y) $$ verifying the assertion in this special case.

More generally, it seems to me that if $X = \text{colim}(X_0 \leftarrow FA \to FB)$ where $(B,A)$ is a cofibration pair (with the colimit taken in topological monoids), and if the assertion is true for $X_0$ then it is also true for $X$ using the above and by noting that (i) function spaces convert pushouts in the domain to pullbacks and (ii) the classifying space functor preserves homotopy pushouts. This would then give the inductive step.

*Added Later:* I'd like to reiterate that I'm really looking for a decent reference.