In his book Classifying spaces and classifying topoi Moerdijk proves that there exists a wieak homotopy equivalence between the classifying topos of the Haefliger space $\Gamma^q$ and that of the monoid $M(\mathbb R^q)$ of smooth embeddings of $\mathbb R^q$ into itself.
From this, he concludes that the corresponding classifying spaces $B\Gamma^q$ and $BM(\mathbb R^q)$ are weakly homotopy equivalent, using
Let $C$ be a topological category. Then there is a weak homotopy equivalence $f:Sh(BC)\to\mathcal B C$, where $\mathcal B C$ represents the classifying topos of $C$.
My problem is passing from $Sh(BC)$ to $BC$. In fact, since $BC=|Nerve(C)|$ has a base of contractible open sets (true?) its homotopy groups coincide with the étale homotopy (pro)groups of $Sh(BC)$. But the definition of weak homotopy equivalence requires the existence of a continuous function between topological spaces inducing those isomorphism in homotopy groups. So I would need some $$g:B\Gamma^q\to BM(\mathbb R^q)$$ inducing isomorphisms in homotopy, and I fail to see how to recover this from the abstract functor between the topoi of sheaves $$Sh(B\Gamma^q)\to\mathcal B \Gamma^q\to \mathcal BM(\mathbb R^q)\to Sh(BM(\mathbb R^q)).$$
Can someone give a hint? Thank you in advance.