Construction 4.3.5.1 (Slice Simplicial Sets). Let $f: K \rightarrow X$ be a morphism of simplicial sets. We define a simplicial set $X_{/f}$ as follows:
For each $n \geq 0$, an $n$-simplex of $X_{/f}$ is a map of simplicial sets $\overline{f}: \Delta ^ n \star K \rightarrow X$ satisfying $\overline{f}|_{K} = f$.
For every nondecreasing function $\alpha : [m] \rightarrow [n]$ in $\operatorname{{\bf \Delta }}$, the associated map
\[ \alpha ^{\ast }: \{ \textnormal{$n$-simplices of $X_{/f}$} \} \rightarrow \{ \textnormal{$m$-simplices of $X_{/f}$} \} \]carries an $n$-simplex $\overline{f}: \Delta ^ n \star K \rightarrow X$ to the composite map
\[ \Delta ^{m} \star K \xrightarrow { \alpha \star \operatorname{id}_ K} \Delta ^{n} \star K \xrightarrow { \overline{f} } X. \]
We will refer to $X_{/f}$ as the slice simplicial set of $X$ over $f$.