N localic coherent implies locally coherent

By assumption, every object \(X \in T\) is covered by objects from \(T_{\leq n-1}\). Let \(X \in T_{\leq n-1}\). By assumption on \(T\), there exists an effective epimorphism \(\bigsqcup_i U_i \twoheadrightarrow X\) where each \(U_i\) is \(n\)-coherent. Because \(X\) is \((n-1)\)-truncated, this map descends to a map \(\bigsqcup_i \tau_{n-1}(U_i) \twoheadrightarrow X\), which is still an effective epimorphism. By Proposition 6.108(1), each \(\tau_{n-1}(U_i)\) is again \(n\)-coherent. We conclude that every \(X \in T\) is covered by objects from \(T_{\leq n-1}^{n\mathrm{-coh}}\).Since \(T\) is \((n+1)\)-coherent, it follows that \(T_{\leq n-1}^{n\mathrm{-coh}}\) is closed under finite limits. By Corollary 6.106, it follows that \(T\) is both coherent and locally coherent.