Variant 3.5.9.10. Suppose we are given a commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr] & & Y \ar [dl] \\ & Z, & } \]
where the vertical maps are Kan fibrations. Then $f$ is $n$-truncated if and only if, for every vertex $z \in Z$, the induced map $f_{z}: X_{z} \rightarrow Y_{z}$ is $n$-truncated. To prove this, we can use Proposition 3.1.7.1 to reduce to the case where $f$ is a Kan fibration. In this case, the desired result follows from the criterion of Proposition 3.5.9.8 (since a Kan complex can be realized as a fiber of $f$ if and only if it can be realized as a fiber of $f_{z}$ for some vertex $z \in Z$).