Exercise 11.6.0.33. See Proposition 1.2.4.7.
Let $0 \leq i \leq n$ be integers. For $j \in [n] \setminus \{ i\} $, we can regard the map $\delta ^{j}_{n}$ of Construction 1.1.1.4 of simplicial sets from $\Delta ^{n-1}$ to the horn $\Lambda ^{n}_{i} \subseteq \Delta ^{n}$. Show that, for any simplicial set $S_{\bullet }$, the construction
determines an injection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_{i}, S_{\bullet } ) \rightarrow \prod _{ j \in [n] \setminus \{ i\} } S_{n-1}$, whose image is the collection of “incomplete” sequences $( \sigma _0, \ldots , \sigma _{i-1}, \bullet , \sigma _{i+1}, \ldots , \sigma _ n)$ satisfying $d^{n-1}_ j(\sigma _ k) = d^{n-1}_{k-1}( \sigma _{j})$ for $j, k \in [n] \setminus \{ i\} $ with $j < k$.