Construction 8.1.2.7. Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, let $\sigma $ be an $n$-simplex of $K^{\operatorname{op}}$, and let $\tau $ be an $n$-simplex of $\operatorname{\mathcal{C}}_{f/}$. Then the composition
can be identified with an $n$-simplex $T(\sigma ,\tau )$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$. The construction $(\sigma , \tau ) \mapsto T(\sigma ,\tau )$ determines a morphism of simplicial sets $K^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$, which we can identify with a map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\operatorname{op}} }( K^{\operatorname{op}}, \operatorname{Tw}(\operatorname{\mathcal{C}}) )$ which we refer to as the coslice inclusion. We will be particularly interested in the special case where $K = \Delta ^0$, so that $f$ can be identified with a vertex $X \in \operatorname{\mathcal{C}}$. In this case, we can identify the coslice inclusion with a morphism of simplicial sets $\iota _ X: \operatorname{\mathcal{C}}_{X/} \rightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$, which we will denote by $\iota _{X}$.