$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Exercise Let $n \geq 0$ be an integer. For $0 \leq j \leq n$, the map $\delta ^{j}: [n-1] \rightarrow [n]$ of Notation determines a map of simplicial sets $\Delta ^{n-1} \rightarrow \Delta ^{n}$ which factors through the simplicial subset $\operatorname{\partial \Delta }^ n \subseteq \Delta ^ n$. We therefore obtain a map of simplicial sets $\Delta ^{n-1} \rightarrow \operatorname{\partial \Delta }^{n}$, which we will also denote by $\delta ^{j}$. Show that, for any simplicial set $S_{\bullet }$, the construction

\[ ( f: \operatorname{\partial \Delta }^{n} \rightarrow S_{\bullet } ) \mapsto \{ f \circ \delta ^{j} \} _{0 \leq j \leq n} \]

determines an injective map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{n}, S_{\bullet } ) \rightarrow \prod _{ j \in [n]} S_{n-1}, \]

whose image is the collection of sequences of $(n-1)$-simplices $(\sigma _0, \sigma _1, \ldots , \sigma _ n)$ satisfying the identities $d_ j(\sigma _ k) = d_{k-1}( \sigma _{j})$ for $0 \leq j < k \leq n$.