Corollary 1.5.5.7. Let $p: X \rightarrow Y$ be a trivial Kan fibration of simplicial sets. Then, for every simplicial set $B$, the induced map $\operatorname{Fun}( B, X ) \rightarrow \operatorname{Fun}( B, Y )$ is a trivial Kan fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Corollary 1.5.5.6 in the special case $A = \emptyset $. $\square$