Remark 6.43.
The object \((\cosk_{n-1} U_{\bullet})_n\) is known as the \(n\)-th matching object of \(U_\bullet\). By the pointwise formula for right Kan extensions, it may be computed as a limit over the category \((\simp^{\leq n-1})_{/[n]}\). Since this category is finite, the object \((\cosk_{n-1} U_\bullet)_n\) is a finite limit of the objects \(U_0, \ldots, U_{n-1}\).