Notation 4.6.6.1. Let $K_{-}$, $K_{+}$, and $\operatorname{\mathcal{C}}$ be simplicial sets, and suppose we are given a morphism $f_{\pm }: K_{-} \star K_{+} \rightarrow \operatorname{\mathcal{C}}$. Set $f_{-} = f_{\pm } |_{ K_{-} }$ and $f_{+} = f_{\pm } |_{ K_{+} }$, and let $\pi : \operatorname{\mathcal{C}}_{ / f_{+} } \rightarrow \operatorname{\mathcal{C}}$ denote the projection map. Then $f_{\pm }$ determines a morphism of simplicial sets $\widetilde{f}_{-}: K_{-} \rightarrow \operatorname{\mathcal{C}}_{ / f_{+} }$ for which the diagram
\[ \xymatrix { & \operatorname{\mathcal{C}}_{ / f_{+} } \ar [dr]_{ \pi } & \\ K_{-} \ar [ur]_{ \widetilde{f}_{-} } \ar [rr]^{ f_{-} } & & \operatorname{\mathcal{C}}} \]
is commutative. In this situation, we let $\operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} }$ denote the coslice simplicial set $( \operatorname{\mathcal{C}}_{ / f_{+} } )_{ \widetilde{f}_{-} / }$.