Proposition 1.5.7.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \hookrightarrow Y$ be an inner anodyne morphism of simplicial sets. Then the induced map $p: \operatorname{Fun}( Y, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( X, \operatorname{\mathcal{C}})$ is a trivial Kan fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. To show that $p$ is a trivial Kan fibration, it will suffice to show that it is weakly right orthogonal to every monomorphism of simplicial sets $f': X' \hookrightarrow Y'$. This is equivalent to the assertion that every map of simplicial sets
\[ g_0: (Y \times X' ) \coprod _{ (X \times X')} (X \times Y' ) \rightarrow \operatorname{\mathcal{C}} \]
can be extended to a map $g: Y \times Y' \rightarrow \operatorname{\mathcal{C}}$. This follows from Proposition 1.5.6.7, since $\operatorname{\mathcal{C}}$ is an $\infty $-category and the map
\[ u_{f,f'}: (Y \times X' ) \coprod _{ (X \times X')} (X \times Y' ) \hookrightarrow Y \times Y' \]
is inner anodyne (Lemma 1.5.7.5). $\square$