$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset, and let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then the simplicial set $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}})$ is an $\infty $-category. Moreover, if the inclusion $A \hookrightarrow B$ is inner anodyne, then $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}})$ is a contractible Kan complex.

Proof. Apply Proposition in the special case $S = \Delta ^{0}$. $\square$