Truncation preserves lower homotopy groups

Choose a left exact localization \(L\colon \PSh(C)\to T\) with fully faithful right adjoint \(R\). Form the \(n\)-truncation \(R(X)\to\tau_n^{\mathrm{pre}}R(X)\) in \(\PSh(C)\). Truncations and homotopy group objects in the presheaf topos are computed pointwise, so classical homotopy theory gives isomorphisms on \(\pi_k\) for \(k\leq n\). Applying \(L\) preserves these homotopy group objects by Lemma 3.25. It also identifies \(L\tau_n^{\mathrm{pre}}R(X)\) with \(\tau_nX\cong Y\) by Lemma 3.8. The resulting isomorphisms are precisely those in the statement. See also [Lurie 2009, Lemma 6.5.1.9].