ScalingStacks

Proof. Let $A$ be an object in $s(s{\operatorname{\mathcal{S}}})$ (i.e., a simplicial object in $s{\operatorname{\mathcal{S}}}$) defined by

$$A(m)=\coprod_{[k_{0}]\rightarrow\dots\rightarrow[k_{m}]\in I}F(k_{0})\in s{\operatorname{\mathcal{S}}}.$$

There is an augmentation map $A(0)\rightarrow\discnerve I$, and the induced map $\diag^{\prime}{A}\rightarrow\discnerve I$ is a Reedy weak equivalence in $s{\operatorname{\mathcal{S}}}$, where $\diag^{\prime}\colon s(s{\operatorname{\mathcal{S}}})\rightarrow s{\operatorname{\mathcal{S}}}$ denotes the prolongation of the diagonal functor, in this case defined by $(\diag^{\prime}A)_{n}\approx\diag\left([m]\rightarrow A(m)_{n}\right)$. The result follows from isomorphisms

$$\Map_{s{\operatorname{\mathcal{S}}}}(\diag^{\prime}{A},W)\approx\Tot(\Map_{s{\operatorname{\mathcal{S}}}}(A({-}),W))\approx\holim_{[k]\rightarrow I\in\boldsymbol{\Delta}^{\operatorname{op}}I}W_{k},$$

and the fact that $\Map_{s{\operatorname{\mathcal{S}}}}(\discnerve I,W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(\diag^{\prime}{A},W)$ is a weak equivalence since $W$ is Reedy fibrant. ∎