Remark 4.6.6.2. In the situation of Notation 4.6.6.1, we can also identify $f_{\pm }$ with a morphism of simplicial sets $\widetilde{f}_{+}: K_{+} \rightarrow \operatorname{\mathcal{C}}_{ f_{-} / }$. Let $Y$ be any simplicial set. Using Proposition 4.3.5.13 we see that the following data are equivalent:
- $(1)$
Morphisms from $Y$ to $( \operatorname{\mathcal{C}}_{ / f_{+} } )_{ \overline{f}_{-} / }$.
- $(2)$
Morphisms from $Y$ to $( \operatorname{\mathcal{C}}_{ f_{-} /} )_{ / \overline{f}_{+} }$.
- $(3)$
Morphisms $f: K_{-} \star Y \star K_{+} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f|_{ K_{-} \star K_{+} } = f_{\pm }$.
It follows that the simplicial set $\operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} } = ( \operatorname{\mathcal{C}}_{ / f_{+} } )_{ \widetilde{f}_{-} / }$ can also be identified with the slice simplicial set $(\operatorname{\mathcal{C}}_{ f_{-} / })_{ /\widetilde{f}_{+} }$.