Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 4.3.5.7. Let $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $f = \operatorname{N}_{\bullet }(F)$ denote the induced morphism of simplicial sets from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. For each $n \geq 0$, we have canonical bijections

\begin{eqnarray*} \{ \textnormal{$n$-simplices of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/f}$} \} & \simeq & \{ \textnormal{Morphisms $\overline{f}: \Delta ^ n \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with $\overline{f}|_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})} = f$} \} \\ & \simeq & \{ \textnormal{Morphisms $\overline{f}: \operatorname{N}_{\bullet }([n]) \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with $\overline{f}|_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})} = f$} \} \\ & \simeq & \{ \textnormal{Morphisms $\overline{f}: \operatorname{N}_{\bullet }([n] \star \operatorname{\mathcal{K}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with $\overline{f}|_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})} = f$} \} \\ & \simeq & \{ \textnormal{Functors $\overline{F}: [n] \star \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ with $\overline{F}|_{\operatorname{\mathcal{K}}} = F$} \} \\ & \simeq & \{ \textnormal{Functors $[n] \rightarrow \operatorname{\mathcal{C}}_{/F}$} \} \\ & \simeq & \{ \textnormal{$n$-simplices of $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/F} )$ } \} . \end{eqnarray*}

Here the third bijection comes from Example 4.3.3.20, the fourth from Proposition 1.2.2.1, and the fifth from Proposition 4.3.2.10. These bijections depend functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore determine an isomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/f} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/F} )$. Similarly, we have a canonical isomorphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{f/} \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{F/} )$. For a more general statement, see Corollary 4.3.5.18.