Lemma 7.5.3.8. Let $\operatorname{\mathcal{C}}$ be a category. Then the collection of projection maps $\{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \} _{ C \in \operatorname{\mathcal{C}}}$ exhibit $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ as the colimit of the diagram
\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}\quad \quad C \mapsto \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ). \]
Proof.
Fix an integer $n \geq 0$; we wish to show that the canonical map
\[ \rho : \varinjlim _{C \in \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{N}_{n}( \operatorname{\mathcal{C}}_{C/} ) \rightarrow \operatorname{N}_{n}( \operatorname{\mathcal{C}}) \]
is an isomorphism in the category of sets. Let $\sigma $ be an $n$-simplex of $\operatorname{N}_{n}(\operatorname{\mathcal{C}})$, given by a diagram
\[ X_0 \rightarrow X_1 \rightarrow \cdots \rightarrow X_ n \]
in the category $\operatorname{\mathcal{C}}$. Then the fiber $\rho ^{-1} \{ \sigma \} $ can be identified with the colimit
\[ \varinjlim _{C \in \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X_0 ), \]
formed in the category of sets. This colimit consists of a single element, represented by the identity morphism $\operatorname{id}_{ X_0} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, X_0)$.
$\square$