Example 4.3.5.15. Let $X$ be a simplicial set containing a vertex $x$. Let $Y$ be a simplicial set, and let $v$ and $v'$ denote the cone points of $Y^{\triangleright }$ and $Y^{\triangleleft }$, respectively. Then Proposition 4.3.5.13 supplies bijections
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, X_{/x} ) \simeq \{ \textnormal{Morphisms $f: Y^{\triangleright } \rightarrow X$ with $f(v) = x$} \} \]
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, X_{x/} ) \simeq \{ \textnormal{Morphisms $f: Y^{\triangleleft } \rightarrow X$ with $f(v') = x$} \} . \]