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Remark 4.3.5.15. Let $f_{-}: K_{-} \rightarrow X$ and $f_{+}: K_{+} \rightarrow X$ be morphisms of simplicial sets. Then Proposition 4.3.5.13 supplies bijections between the following:

$(1)$

The collection of morphisms $\overline{f}_{-}: K_{-} \rightarrow X_{/ f_{+}}$ for which the composition $K_{-} \xrightarrow { \overline{f}_{-} } X_{/f_{+}} \rightarrow X$ is equal to $f_{-}$.

$(2)$

The collection of morphisms $\overline{f}_{+}: K_{+} \rightarrow X_{f_{-} /}$ for which the composition $K_{-} \xrightarrow { \overline{f}_{+}} X_{f_{-}/} \rightarrow X$ is equal to $f_{+}$.

$(3)$

The collection of morphisms $f_{\pm }: K_{-} \star K_{+} \rightarrow X$ for which $f_{\pm }|_{K_{-}} = f_{-}$ and $f_{\pm }|_{ K_{+}} = f_{+}$.

Suppose we are given a morphism of simplicial sets $f_{\pm }: K_{-} \star K_{+} \rightarrow X$ as in $(3)$, corresponding to morphisms $\overline{f}_{-}: K_{-} \rightarrow X_{ / f_{+} }$ and $\overline{f}_{+}: K_{+} \rightarrow X_{f_{-} /}$ as in $(1)$ and $(2)$, respectively. For every simplicial set $Y$, Proposition 4.3.5.13 supplies canonical bijections between the following:

$(1')$

The collection of morphisms $Y \rightarrow ( X_{ / f_{+} } )_{ \overline{f}_{-} / }$.

$(2')$

The collection of morphisms $Y \rightarrow ( X_{ f_{-} /} )_{ / \overline{f}_{+} }$.

$(3')$

The collection of morphisms $f: K_{-} \star Y \star K_{+} \rightarrow X$ satisfying $f|_{ K_{-} \star K_{+} } = f_{\pm }$.

These bijections determine a canonical isomorphism of simplicial sets

\[ (X_{f/} )_{ / \overline{f}' } \simeq ( X_{/ f'} )_{ \overline{f} / }. \]

We will henceforth abuse notation by denoting either of these simplicial sets by $X_{ f_{-} / \, / f_{+} }$. Beware that the simplicial set $X_{ f_{-} / \, / f_{+} }$ depends not only on $f_{-}$ and $f_{+}$, but also on their common extension $f_{\pm }: K_{-} \star K_{+} \rightarrow X$.