Corollary 3.3.1.9. Let $S_{\bullet }$ be a semisimplicial set. Then, for every simplicial set $Y_{\bullet }$, composition with the map $\iota : S_{\bullet } \hookrightarrow S^{+}_{\bullet }$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Morphisms of simplicial sets $f: S^{+}_{\bullet } \rightarrow Y_{\bullet }$} \} \ar [d] \\ \{ \textnormal{Morphisms of semisimplicial sets $f_0: S_{\bullet } \rightarrow Y_{\bullet }$} \} . } \]