ScalingStacks

Proof of statement (2). By (13.5) the natural maps $W\rightarrow W^{E(m)}$ are categorical equivalences, and hence weak equivalences in the complete Segal space model category structure by (13.6). Thus the induced map on homotopy colimits

$$W=\diag^{\prime}([m]\mapsto W)\rightarrow\diag^{\prime}([m]\mapsto W^{E(m)})=\widetilde{W}$$

is a weak equivalence in the complete Segal space model category structure. ∎