Corollary 3.10.
For a morphism \(f\colon X \to Y\) in a topos \(T\), the pullback functor \(f^*\colon T_{/Y} \to T_{/X}\) preserves \(n\)-truncated objects and commutes with \(n\)-truncation: for a map \(Z \to Y\) we have \(\tau_n(f^*Z/X) \cong f^*\tau_{n}(Z/Y)\).
Proof
Lemma 3.8 applies: the functor \(f^*\) preserves limits (as it has a left adjoint) and preserves colimits (by universality of colimits), so that it admits a right adjoint \(f_*\colon T_{/X} \to T_{/Y}\) by the adjoint functor theorem.