I'm trying to understand the proof of Thm 4.16 in B. Conrad's notes on Cohomological Descent. A special case of this theorem is the following form:
If $k$ is a field of characteristic zero and $S$ is a proper $k$-scheme, there exists a proper hypercover $X_\bullet \to S$ such that each $X_n$ is a smooth proper $k$-scheme. Inductively one proves that there exists a $n$-truncated proper hypercover $X_{\leq n}$, where the induction basis $n=0$ is the resolution of singularity. Assuming we have a solution $X_{\leq n}$, then we take $\text{sk}_{n+1} \text{cosk}_{n} X_{\leq n}$, and a resolution of singularity$$X_{n+1}'\to \left(\text{sk}_{n+1} \text{cosk}_{n} X_{\leq n}\right)_{n+1}.$$ The part which I don't understand is how we get a new $(n+1)$-truncated hypercover out of this: we are need degeneracy maps. In low degrees, by which I mean if we do $\text{cosk}_0$, the we can just take $$X_1:=X_1'\coprod X_0$$ but I don't understand how (or what) we do for higher cases. Conrad applies the construction of the proof of the existence of a split hypercover (Thm 4.13 in his notes), but I don't understand how we can apply this as we don't have a hypercover to begin with and I also don't see how that yields what we want... (to be clear I'm sure this works and I just don't understand Conrad's notes, nothing more).