Corollary 3.5.2.13. Let $m$ and $n$ be integers, let $B$ be a simplicial set of dimension $\leq m$, and let $f: X \rightarrow Y$ be a morphism of Kan complexes which is $(m+n)$-connective. Then the induced map $\operatorname{Fun}(B, X) \rightarrow \operatorname{Fun}(B,Y)$ is $n$-connective.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Applying Proposition 3.5.2.11 in the special case $A = \emptyset $. $\square$