Lemma 3.6.
A morphism \(f\colon X \to Y\) is \(n\)-truncated if and only if the induced map of animae \(f_*\colon \Hom_C(Z,X) \to \Hom_C(Z,Y)\) is \(n\)-truncated for all \(Z \in C\).
Proof
Immediate from Yoneda lemma and the fact that \(\Hom_C(Z,-)\) preserves pullbacks.