It is claimed in Rezk's notes, Prop 15.24, pg. 46, that we have an obvious map of push out
What I don't understand is the explicit top horizontal map: what is it?
It seems that he is using the earlier result
Thoughts:
Fix a $n$. Then for each $f \in (\partial \Delta^k)_n$ we may decompose it as a surjective morphism $f:[n] \rightarrow S \rightarrow [k]$, where $S \subsetneq k$. We construct a bijetion of the unique linearly ordered set $[j] \simeq S$.
Hence, each $f$ induces a unique map $f' \in Hom_{\Delta^{surj}}([n],[j])$ and we send $(a,f) \mapsto ((f')^*a , f')$. Where $(f')^* a \in X^{nd}_j$.
Note that the construction of $Sk$ is functorial. The element $a\in X_k^{nd}$ regarded as a morphism $\Delta^k\to X$ via the Yoneda lemma induces a map $\partial\Delta^k = Sk_{k-1}(\Delta^k)\to Sk_{k-1}(X)$.