Kerodon

Remark 4.3.5.2. Let $f: K \rightarrow X$ be a morphism of simplicial sets, and let $\overline{f}: \Delta ^ n \star K \rightarrow X$ be an $n$-simplex of the slice simplicial set $X_{/f}$. Then the restriction $\overline{f}|_{\Delta ^ n}$ is an $n$-simplex of $X$. The construction $\overline{f} \mapsto \overline{f}|_{\Delta ^ n}$ determines a morphism of simplicial sets $X_{/f} \rightarrow X$, which we will refer to as the projection map or the forgetful functor (in the case where $X$ is an $\infty $-category). We will often abuse notation by identifying a vertex of $X_{/f}$ with its image in $X$.