Corollary 4.3.5.14. Let $K$ be a simplicial set. Then the join functor
\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{K/} \quad \quad Y \mapsto Y \star K \]
admits a right adjoint, given on objects by the slice construction $(f: K \rightarrow X) \mapsto X_{/f}$. Similarly, the join functor
\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{K/} \quad \quad Y \mapsto K \star Y \]
admits a right adjoint, given on objects by the coslice construction $(f: K \rightarrow X) \mapsto X_{f/}$.