ScalingStacks

Proof. Recall from (7.2) that $f$ is a weak equivalence in the complete Segal space model category if and only if $\Map_{s{\operatorname{\mathcal{S}}}}(f,W)$ is a weak equivalence of spaces for each complete Segal space $W$. This is equivalent to supposing that $W^{f}\colon W^{V}\rightarrow W^{U}$ is a Reedy weak equivalence for each complete Segal space $W$, since $(W^{f})_{n}\approx\Map_{s{\operatorname{\mathcal{S}}}}(f,W^{F(n)})$ and since $W^{F(n)}$ is a complete Segal space by (7.3). The result now follows by noting that $W^{f}$ is a categorical equivalence between complete Segal spaces by (13.5) and (7.3), and thus is a Reedy weak equivalence by (13.4). ∎