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Remark 4.6.4.8. Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, which we identify with a vertex of the simplicial set $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. For any simplicial set $J$, we have canonical isomorphisms

\[ \operatorname{Fun}(J, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} ) \simeq \operatorname{Fun}_{ K/ }( J \diamond K, \operatorname{\mathcal{C}}) \quad \quad \operatorname{Fun}(J, \{ F\} \operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}_{ K/ }( K \diamond J, \operatorname{\mathcal{C}}), \]

where $J \diamond K$ and $K \diamond J$ denote the blunt joins introduced in Notation 4.5.8.3. Restricting to vertices, we obtain bijections

\[ \{ \textnormal{Morphisms $J \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} $} \} \simeq \{ \textnormal{Morphisms $\overline{F}: J \diamond K \rightarrow \operatorname{\mathcal{C}}$ with $\overline{F}|_{K} = F$} \} \]
\[ \{ \textnormal{Morphisms $J \rightarrow \{ F\} \operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \operatorname{\mathcal{C}}$} \} \simeq \{ \textnormal{Morphisms $\overline{F}': K \diamond J \rightarrow \operatorname{\mathcal{C}}$ with $\overline{F}'|_{K} = F$} \} . \]