ScalingStacks

Proof. By [52, A.3.4.15], we can choose a small subcategory ${\mathcal{V}}\subset{\mathcal{A}}$ which contains ${\mathcal{C}}$ and such that ${\mathcal{V}}$ is an $(\Ar[n])$-chunk for each $n$ and moreover $\mathrm{N}(({\mathcal{C}})^{\cf})$ is equivalent to $\mathrm{N}(({\mathcal{V}})^{\cf})$. Then as discussed above, [52, 4.2.4.4] implies that for each $n$ the natural map

$$\mathrm{N}((({\mathcal{V}})^{\Ar[n]})^{\cf})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}(\mathrm{N}(\Ar[n]),\mathrm{N}(({\mathcal{V}})^{\cf}))$$

is a categorical equivalence of simplicial sets. ∎