Exercise 8.1.1.8 (Slices of Twisted Arrows). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $f: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{C}}$, which we regard as a vertex of the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$. Show that there is a canonical isomorphism of simplicial sets
\[ \operatorname{Tw}(\operatorname{\mathcal{C}})_{ / f } \simeq \operatorname{Tw}( \operatorname{\mathcal{C}}_{ X/ \, / Y} ). \]
Here $\operatorname{\mathcal{C}}_{X/ \, /Y}$ denotes the simplicial set $(\operatorname{\mathcal{C}}_{X/ })_{/Y} \simeq ( \operatorname{\mathcal{C}}_{/Y} )_{X/}$, obtained either by promoting $Y$ to a vertex of $\operatorname{\mathcal{C}}_{X/}$ or $X$ to a vertex of $\operatorname{\mathcal{C}}_{/Y}$ by means of the edge $f$ (see Remark 4.6.6.2).