ScalingStacks

Proof. We must show that for each $m,n\geq 0$ the natural maps $N(D^{C},\we(D^{C}))\rightarrow N(D,W)^{NC}\rightarrow N(D,W)^{\discnerve C}$ induce one-to-one correspondences amongst the sets of

1. (1)

   functors $[m]\times[n]\rightarrow D^{C}$ which carry “vertical” maps into $\we(D^{C})$,

2. (2)

   maps $F(m)\times\Delta[n]\rightarrow N(D,W)^{NC}$ of simplicial spaces, and

3. (3)

   maps $F(m)\times\Delta[n]\rightarrow N(D,W)^{\discnerve C}$ of simplicial spaces.

By (3.8) and (3.12) it will suffice to show this in the case $m=n=0$, in which case the result becomes a straightforward computation. ∎