ScalingStacks

Let $S^{\prime\prime}$ be a subset of $S$ with $n$ elements, and let $T$ be a finite subset of $M$. Let $a:\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S^{\prime\prime},\{\xi(1),\ldots\xi(n)\})\wedge\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\setminus S^{\prime\prime},T)\to L^{\bullet}(T,S,e^{n})$ be the map of Lemma 8.1.2. Let $\alpha\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S^{\prime\prime},\{\xi(1),\ldots\xi(n)\})$ and $\beta\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\setminus S^{\prime\prime},T)$. Let $\theta=\alpha\boxtimes\beta=a(\alpha\wedge\beta)$. The statement (8.1.2) will follow from the following property:

(8.1.3)

$$a(d(\alpha\otimes\beta))=d(\theta).$$