this we take the diagonal so the collection of $n$-simplices is $S(\mathbb{A})_{n,n}$, and by noticing that the result has a constant simplicial set of objects, then apply the lemma.

The Dwyer-Kan ‘simplicial groupoid’ functor.
--------------------------------------------

Let $K$ be a simplicial set. Near the start of simplicial homotopy theory, Kan showed how, if $K$ was reduced (that is, if $K_0$ was a singleton), then the free group functor applied to $K$ in a subtle way, gave a simplicial group whose homotopy groups were those of $K$, with a shift of dimension. With Dwyer in , he gave the necessary variant of that construction to enable it to apply to the non-reduced case. This gives a ‘simplicial groupoid’ $G(K)$ as follows:

The object set of all the groupoids $G(K)_n$ will be in bijective correspondence with the set of vertices $K_0$ of $K$. Explicitly this object set will be written $\{\overline{x} ~|~ x \in K_0\}$.

The groupoid $G(K)_n$ is generated by edges $$\overline{y} : \overline{d_1d_2 \ldots d_{n+1} y} \to \overline{d_0d_2\ldots d_{n+1}y} \quad \mbox{ for } y \in K_{n+1}$$with relations $\overline{s_0x} = id_{\overline{d_1d_2 \ldots d_n x}}$. Note since these just ‘kill’ some of the generating edges, the resulting groupoid $G(K)_n$ is still a free 