Lemma 3.9.
For every \(n\geq -2\), the truncation functor \(\tau_n\colon T\to T_{\leq n}\) of a topos preserves finite products.
Proof
Choose a left exact localization \(L\colon \PSh(C)\to T\) with fully faithful right adjoint \(R\). In a presheaf topos, truncation is computed pointwise, and truncation of animae preserves finite products. It follows that \(\tau_n^{\mathrm{pre}}\) preserves finite products in \(\PSh(C)\). By Lemma 3.8, the truncation functor on \(T\) is naturally isomorphic to the composite \(L\tau_n^{\mathrm{pre}}R\). The functors \(L\) and \(R\) preserve finite limits, so this composite preserves finite products.