ScalingStacks

Proof. By Theorem 4.7, we know that $\mathcal{T}r(\qc(X))$, which is a tensor product, is also calculated by a fiber product

$$\mathcal{T}r(\qc(X))\simeq\qc(X\times_{X\times X}X).$$

On the other hand, by Corollary 4.10, we know that $\mathcal{Z}(\qc(X))$, which consists of functors, is also calculated by a tensor product

$$\mathcal{Z}(\qc(X))\simeq\mathcal{T}r(\qc(X)).$$

∎