Lemma 3.8.
Consider a functor \(F\colon C \to D\).
If \(F\) preserves pullbacks, then it preserves \(n\)-truncated objects for all \(n \geq -2\).
If \(F\) furthermore admits a right adjoint, and \(C\) and \(D\) admit \(n\)-truncation functors, then \(F\) commutes with \(n\)-truncation: for any \(X \in C\) the canonical map
\[\tau_n(F(X)) \to F(\tau_n X)\]is an isomorphism.
Proof
Part (1) is clear, since \(n\)-truncatedness is formulated in terms of pullbacks.For part (2), the claim is equivalent to the claim that the right adjoint \(G\colon D \to C\) of \(F\) preserves \(n\)-truncated objects. This is an instance of (1).