ScalingStacks

Proof. Being local with respect $\mathrm{Segal}_{\Delta^{\!\times n}}$ and $\mathrm{Glob}_{\Delta^{\!\times n}}$ (or to $\mathrm{Segal}_{\Theta_{n}}$ for $\Theta_{n}$-presheaves) implies that the presheaf is the nerve of a strict $n$-category. Such an $n$-category is gaunt if and only if it is local with respect to the morphisms $\sigma^{k}(E)\to\sigma^{k}(C_{0})$. This last follows from locality with respect to $\mathrm{Comp}_{\Delta^{\!\times n}}$ (or, respectively, with respect to $\mathrm{Comp}_{\Theta_{n}}$) because the square

is a pushout square of strict $n$-categories. ∎