In Higher Topos Theory Lurie gives the following description of a cosimplicial simplicial space $Q^\bullet$ which geometric realization gives the straightening over a point: For $n \geq 0$ let $P_{[n]}$ be the partial order of nonempty subsets of $[n]=\lbrace 0,...,n\rbrace$ and $K_{[n]}$ its nerve. This should be isomorphic to the cube $ (\Delta^1)^{n+1}$ missing its $(0,...,0)$ vertex. Then we obtain $Q^n$ by collapsing for $0 \leq i \leq n$ the subset $$ (\Delta^1)^{(j:0 \leq j <i)} \times \lbrace 1 \rbrace \times (\Delta^1)^{(i< j \leq n)} $$ to its quotient $(\Delta^1)^{(i< j \leq n)}$. The maps for the copresheaf-structure are induced by the obvious map on partially ordered sets( for $f:[n] \to [m]$, map a subset $I$ to $f(I)$). Lurie then states that (after geometric realization) these $Q^n$ are isomorphic to standard simplices and the copresheaf maps are even compatible with the usual facemaps of the copresheaf $\Delta^n$, but that the degeneracies differ. As an example he states the product of degeneracies $Q^n \to (Q^1)^n$ not to be injective (after geometric realization). Indeed when I attempt to calculate these for low dimensions (n=1,2) it seems to yield the standard simplices, but furthermore also with the standard degeneracies.
So my question is: Can somebody help me with an explicit description of $Q^n$ and its degeneracy maps in low dimensions and how it differs from $\Delta^n$. I am also confused why Lurie applies geometric realization before making his remark about the degeneracies. Any help will be greatly appreciated.
Edit: I will spell out some of my reasoning so that it may become clearer where i'm confused: There is a clear map from $Q^\bullet \to \Delta^\bullet$ induced by sending a subset to its largest member. But notice that for any subset its highest member should be all that counts after the identifications. Two subsets with a number $j$ as their highest member correspond to vertices of the cube with $1$ in $j$th position and $0$ from there on. But these get identified. Furthermore any map into a simplex should be determined by its $0$-skeleton, as simplexes admit unique fillers. So why is the map not an isomorphism of cosimplicial sets?