The augmented Čech nerve

Given a morphism \(f\colon U \to X\), we will write \(\check{C}^+_{\bullet}(f)\) for the augmented simplicial diagram \(j_*(f) \colon \simp_+\catop \to C\), and refer to this as the augmented Čech nerve of \(f\). Under the canonical equivalence between \(\simp_+\catop\) and \((\simp\catop)^{\triangleright}\), we may think of \(\check{C}^+_{\bullet}(f)\) as encoding a cocone on \(\check{C}_{\bullet}(f)\) with cone point \(X\).