Proposition 4.3.5.13 (0189)

Proposition 4.3.5.13. Let $f: K \rightarrow X$ be a morphism of simplicial sets, and let $c: X_{/f} \star K \rightarrow X$ be the slice contraction morphism of Construction 4.3.5.12. Then, for any simplicial set $Y$, postcomposition with $c$ induces a bijection

\[ \theta _{Y}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, X_{/f} ) \rightarrow \{ \textnormal{Morphisms $\overline{f}: Y \star K \rightarrow X$ satisfying $\overline{f}|_{K} = f$} \} \]

Similarly, postcomposition with the coslice contraction morphism $c': K \star X_{f/} \rightarrow X$ induces a bijection

\[ \theta '_{Y}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, X_{f/} ) \rightarrow \{ \textnormal{Morphisms $\overline{f}: K \star Y \rightarrow X$ satisfying $\overline{f}|_{K} = f$} \} . \]

Proof. In the case where $Y$ is a standard simplex, both assertions follow immediately from the definition of the simplicial sets $X_{/f}$ and $X_{f/}$. Since every simplicial set can be realized as a colimit of simplices (Remark 1.1.3.13), it will suffice to show that the constructions $Y \mapsto \theta _{Y}$ and $Y \mapsto \theta '_{Y}$ carry colimits of simplicial sets to limits in the arrow category $\operatorname{Fun}([1], \operatorname{Set})$. This follows from the observation that the functors

\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{K/} \quad \quad Y \mapsto Y \star K, Y \mapsto K \star Y \]

preserve small colimits (see Remark 4.3.3.31). $\square$