Proposition 1.1.0.12. Let $n$ be a nonnegative integer and regard the identity map $\operatorname{id}_{[n]}: [n] \rightarrow [n]$ as an $n$-simplex of $\Delta ^ n$. For every simplicial set $S_{\bullet }$, evaluation on $\operatorname{id}_{[n]}$ induces a bijection
\[ \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \Delta ^ n, S_{\bullet } ) \rightarrow S_{n} \quad \quad f \mapsto f( \operatorname{id}_{[n] } ). \]