Corollary 3.5.2.12. Let $m$ and $n$ be integers, let $B$ be a simplicial set of dimension $\leq m$, and let $X$ be a Kan complex which is $(m+n)$-connective. Then, for every simplicial subset $A \subseteq B$, the restriction map $\operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X)$ is $n$-connective.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Proposition 3.5.2.11 in the special case $Y = \Delta ^0$. $\square$