Čech nerve is hypercover

Let \(f\colon X \twoheadrightarrow Y\) be an effective epimorphism in \(T\). Then the Čech nerve \(\check{C}(f)\), viewed as a simplicial object in \(T_{/Y}\), is a hypercover of \(Y\). Its matching map in degree zero is \(f\), hence is an effective epimorphism. By Lemma 2.20, the simplicial object \(\check{C}(f)\) is \(0\)-coskeletal, so its matching maps in every positive degree are isomorphisms.