Lemma 2.26.
Let \(C\) be a category with pullbacks. Then the Čech nerve \(\check{C}_{\bullet}(f)\) of any morphism \(f\) is a groupoid object.
Proof
This is immediate from Lemma 2.20, since the augmented Čech nerve \(\check{C}^+_{\bullet}(f)\) of \(f\) is by definition right Kan extended from \((\simp_+^{\leq 0})\catop \subseteq \simp\catop_+\).