ScalingStacks

Proof. The maps $W^{i_{0}},W^{i_{1}}\colon W^{E}\rightarrow W$ are Reedy trivial fibrations if $W$ is a complete Segal space. This is because of parts (2) and (3) of (6.4), together with the observation that

$$(W^{E})_{n}\approx\Map_{s{\operatorname{\mathcal{S}}}}(E,W^{F(n)})\approx(W^{F(n)})_{0}$$

since $W^{F(n)}$ is a complete Segal space by (7.3). Thus, categorically homotopic maps coincide in the Reedy homotopy category, and hence are simplicially homotopic since $W$ is Reedy fibrant. ∎