Example 4.6.4.9. Let $J$ and $K$ be simplicial sets, and let $J \diamond K$ denote the blunt join introduced in Notation 4.5.8.3. For every simplicial set $\operatorname{\mathcal{C}}$, we have a canonical isomorphism
\[ \operatorname{Fun}( J \diamond K, \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}(J, \operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}(J \times K, \operatorname{\mathcal{C}}) } \operatorname{Fun}( K, \operatorname{\mathcal{C}}). \]
Recall that the comparison map $c: J \diamond K \rightarrow J \star K$ of Notation 4.5.8.3 is a categorical equivalence of simplicial sets (Theorem 4.5.8.8). Consequently, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then precomposition with $c$ induces an equivalence of $\infty $-categories
\[ \operatorname{Fun}( J \star K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( J, \operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}(J \times K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}}). \]