# Kerodon

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Example 4.3.5.9. Let $K$ be a simplicial set, let $Y$ be a topological space, and let $f: K \rightarrow \operatorname{Sing}_{\bullet }(Y)$ be a morphism of simplicial sets, which we will identify with a continuous function $F: |K| \rightarrow Y$. For each $n \geq 0$, we have canonical bijections

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

Here the third bijection is provided by Proposition 4.3.4.11. Using the fact that these bijections depend functorially on $[n] \in \operatorname{{\bf \Delta }}$ and invoking the universal property $| \Delta ^ n | \star |K|$ (see Remark 4.3.4.3), we obtain an isomorphism of $\operatorname{Sing}_{\bullet }(Y)_{/f}$ with the iterated fiber product

$\operatorname{Sing}_{\bullet }(Y) \times _{ \operatorname{Fun}( \{ 0\} \times K, \operatorname{Sing}_{\bullet }(Y) ) } \operatorname{Fun}( \Delta ^1 \times K, \operatorname{Sing}_{\bullet }(Y) ) \times _{ \operatorname{Fun}( \{ 1\} \times K, \operatorname{Sing}_{\bullet }(Y) )} \{ f \} .$