ScalingStacks

Proof. Let $\Sigma_{0}$ be the $0$-simplices of $\Sigma$. Since $X$ has affine diagonal, the natural projection $X^{\Sigma}\to X^{\Sigma_{0}}$ is affine. By Proposition 3.24, the product $X^{\Sigma_{0}}$ is perfect, and so by Proposition 3.21 (in the basic case of an affine morphism), the assertion follows. ∎