Corollary 4.4.5.6. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories. For every simplicial set $B$, the induced map $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ is also an isofibration of $\infty $-categories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Proposition 4.4.5.1 in the special case $A = \emptyset $. $\square$