Theorem 8.5.9.5. The partial idempotent $\iota : \operatorname{N}_{\leq 2}( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ of Notation 8.5.9.4 is a categorical equivalence of simplicial sets.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Theorem 8.5.9.5 from Proposition 8.5.9.11. We wish to show that the partial idempotent
\[ \iota : \operatorname{N}_{\leq 2}( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( B \operatorname{N}_{\bullet }(\mathrm{Dy} ) ) \]
is a categorical equivalence of simplicial sets. Since the category $\mathrm{Dy}$ is a groupoid, the simplicial monoid $\operatorname{N}_{\bullet }( \mathrm{Dy} )$ is a Kan complex (Proposition 1.3.5.2). It follows that the simplicial category $B \operatorname{N}_{\bullet }( \mathrm{Dy} )$ is locally Kan. Invoking Theorem 
, we are reduced to showing that $\iota $ induces a weak equivalence of simplicial categories $F: \operatorname{Path}[ \operatorname{N}_{\leq 2}(\operatorname{Idem}) ]_{\bullet } \rightarrow B \operatorname{N}_{\bullet }( \mathrm{Dy} )$, in the sense of Definition 4.6.8.7. Write $X$ for the unique object of the path category $\operatorname{Path}[ \operatorname{N}_{\leq 2}(\operatorname{Idem}) ]_{\bullet }$, so that $F(X)$ is the unique object of $B \operatorname{N}_{\bullet }( \mathrm{Dy} )$. We are then reduced to showing that $F$ induces a weak homotopy equivalence of simplicial monoids
\[ \varphi : E_{\bullet } = \operatorname{Hom}_{ \operatorname{Path}[\operatorname{N}_{\leq 2}(\operatorname{Idem})]}( X, X)_{\bullet } \rightarrow \operatorname{Hom}_{ B \operatorname{N}_{\bullet }( \mathrm{Dy} )}( F(X), F(X) ) = \operatorname{N}_{\bullet }( \mathrm{Dy} ), \]
which follows from Proposition 8.5.9.11. $\square$