Corollary 3.5.9.12. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n$ be an integer. The following conditions are equivalent:
The morphism $f$ is $n$-truncated.
For every morphism of Kan complexes $Y' \rightarrow Y$, the projection map $Y' \times _{Y}^{\mathrm{h}} X \rightarrow Y'$ is $n$-truncated.
For every vertex $y \in Y$, the homotopy fiber $\{ y\} \times ^{\mathrm{h}}_{Y} X$ is $n$-truncated.
Proof. Using Proposition 3.4.0.9, we can reduce to the case where $f$ is a Kan fibration. In this case, we can use Proposition 3.4.0.7 to reformulate conditions $(2)$ and $(3)$ as follows:
For every morphism of Kan complexes $Y' \rightarrow Y$, the projection map $Y' \times _{Y} X \rightarrow Y'$ is $n$-truncated.
For every vertex $y \in Y$, the fiber $\{ y\} \times _{Y} X$ is $n$-truncated.
The equivalence $(1) \Leftrightarrow (3')$ now follows from Proposition 3.5.9.8, and the equivalence $(1) \Leftrightarrow (2')$ from Corollary 3.5.9.11. $\square$