Equivalence between the Dwyer-Kan loop groupoid and the fundamental groupoid Let $X$ be a homotopy 1-type (a space with vanishing homotopy groups above degree one). It is a classical fact that $X$ can be recovered completely from its fundamental groupoid. 
On the the other hand, there is an equivalence of categories between (unpointed) simplicial sets and simplicial groupoids which sends a simplicial set to its Dwyer-Kan loop groupoid: https://ncatlab.org/nlab/show/Dwyer-Kan+loop+groupoid 
When $X$ is a homotopy 1-type, is the Dwyer-Kan loop groupoid equivalent to the image of the fundamental groupoid under the inclusion of groupoids in simpicial groupoids? 
 A: The answer is yes. This follows from checking that the Quillen equivalence $G: sSet ^\to_\leftarrow sGpd : \bar W$ corresponds when we pass to spaces to the expected equivalence between spaces and groupoids in spaces. Then because every $X \in sSet$ is cofibrant and $G$ is left Quillen, $GX$ is equivalent to what we expect, namely the path groupoid of $X$ (well, this is not strictly a groupoid, but there should be an equivalent thing which is a groupoid using Moore paths). Then you can reason as in spaces to conclude that $G$ restricts to an equivalence between 1-types and discrete groupoids.
To see that the equivalence is the expected one, consider the right adjoint $\bar W$. From the description in 3.2 of Dwyer and Kan's paper, one sees that $\bar W$ just takes the diagonal of a simplicial groupoid, regarded via the nerve as a bisimplicial set. This is equivalently the homotopy colimit of the corresponding functor $\Delta^{op} \to sSet$, which is indeed what we expect this adjunction to be doing. 
