ScalingStacks

Proof. We define $\mathcal{Z}_{X}$ on objects by assigning $\mathcal{Z}_{X}(m)=\qc((\mathcal{L}X)^{m})$.

To define $\mathcal{Z}_{X}$ on morphisms, observe first that since $X$ is perfect, $(\mathcal{L}X)^{m}$ is perfect, and for $\Sigma\in 2{\rm Cob}^{\circ}(m,n)$, the mapping stack $X^{\Sigma}=\Map(\Sigma,X)$ is also perfect (special cases of Corollary 3.25). Now for each $\Sigma\in 2{\rm Cob}^{\circ}(m,n)$, consider the correspondence

Since all of the stacks involved are perfect, pullback and pushforward of quasi-coherent sheaves along this correspondence defines a colimit-preserving functor

$$\mathcal{Z}_{X}(\Sigma):\qc((\mathcal{L}X)^{m})\to\qc((\mathcal{L}X)^{n}).$$

Thus applying this construction in families, we obtain a map of spaces

$$\mathcal{Z}_{X}(m,n):2{\rm Cob}^{\circ}(m,n)\to\Fun(\qc((\mathcal{L}X)^{m}),\qc((\mathcal{L}X)^{n})).$$

Suppose now that $\Sigma=\Sigma_{1}\coprod_{\coprod_{k}{S^{1}}}\Sigma_{2}\in 2{\rm Cob}(m,n)$ is obtained by sewing two surfaces $\Sigma_{1}\in 2{\rm Cob}(m,k)$ and $\Sigma_{2}\in 2{\rm Cob}(k,n)$. Then we have a diagram of correspondences

Since all of the stacks involved are perfect, base change provides a canonical equivalence of functors

$$\mathcal{Z}_{X}(\Sigma)\simeq\mathcal{Z}_{X}(\Sigma_{2})\circ\mathcal{Z}_{X}(\Sigma_{1}):\qc((\mathcal{L}X)^{m})\to\qc((\mathcal{L}X)^{n}).$$

Similar diagrams define the higher compositions.

To complete the construction, note that $\mathcal{Z}_{X}$ comes equipped with a canonical symmetric monoidal structure. Namely, by Theorem 4.7, there is a canonical equivalence

$$\mathcal{Z}_{X}(m)=\qc((\mathcal{L}X)^{m})\simeq\qc(\mathcal{L}X)^{\otimes m}=\mathcal{Z}_{X}(1)^{\otimes m},$$

and it clearly extends to a symmetric monoidal structure. ∎