Corollary 6.109.
Consider the unique factorization of \(f\colon X \to Y\) as
\[X \to \tau_{n-1}(X/Y) \to Y\]
into an \((n-1)\)-connected map followed by an \((n-1)\)-truncated map. If \(X\) is \(n\)-coherent and \(Y\) is \(m\)-coherent, then \(\tau_{n-1}(X/Y)\) is \(\max(n,m)\)-coherent.
Proof
By Proposition 6.108(1), the object \(\tau_{n-1}(X/Y)\) is \(n\)-coherent, so we are done if \(n \geq m\). If \(m > n\), we may inductively show that \(\tau_{n-1}(X/Y)\) is \((n + k)\)-coherent for all \(k = 0, \ldots, m-n\). If we already know it is \((n+k-1)\)-coherent, then using that \(Y\) is \(m\)-coherent and \(\tau_{n-1}(X/Y) \to Y\) is \((n-1)\)-truncated, we get by Proposition 6.108(3) that \(\tau_{n-1}(X/Y)\) is \((n+k)\)-coherent.