Kerodon

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Corollary 3.5.9.27. Let $f: X \rightarrow Y$ be an $n$-truncated morphism between Kan complexes. Then, for every simplicial set $B$, the induced map $\operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(B,Y)$ is $n$-truncated.

Proof. Using Proposition 3.1.7.1, we can reduce to the case where $f$ is a Kan fibration. In this case, the desired result follows by applying Proposition 3.5.9.24 in the special case $A = \emptyset $ (and the integer $k$ is equal to zero). $\square$