ScalingStacks

Proof of Theorem 9.1. Suppose that $(\mathcal{C},f)$ and $(\mathcal{D},g)$ are each theories of $(\infty,n)$-categories; that is, they each satisfy axioms C.1-5. By the versality axiom C.5 we have left adjoints

$$L_{1}\colon\mathcal{C}\to\mathcal{D}\qquad\textrm{and}\qquad L_{2}\colon\mathcal{D}\to\mathcal{C}$$

and natural transformations $\eta_{1}\colon L_{1}\circ f\to g$ and $\eta_{2}\colon L_{2}\circ g\to f$ such that $\eta_{i}|_{\mathbb{G}_{n}}$ is an equivalence. Then the theorem follows provided that we demonstrate that both $L_{1}\circ L_{2}$ and $L_{2}\circ L_{1}$ are autoequivalences. We will show this for $L_{2}\circ L_{1}$. The argument for $L_{1}\circ L_{2}$ is identical.

Thus $E\mathrel{\mathop{:}}=L_{2}\circ L_{1}:\mathcal{C}\to\mathcal{C}$ is a colimit preserving endofunctor along with a natural transformation $\eta_{2}\circ L_{2}(\eta_{1})\colon E\circ f\to f$. Since $f$ is dense, Lemma 9.2 ensures that there is a natural transformation $\eta\colon E\to\id$ whose composition with $f$ is $\eta_{2}\circ L_{2}(\eta_{1})$. To see that $\eta$ is an equivalence, let $\mathcal{E}\subseteq\mathcal{C}$ be the full subcategory spanned by those $X$ such that $\eta_{X}$ is an equivalence. Since $E$ preserves colimits, $\mathcal{E}$ is stable under colimits, and since $\eta_{2}\circ L_{2}(\eta_{1})|_{\mathbb{G}_{n}}$ is an equivalence, it follows from (C.2) that $\mathcal{E}=\mathcal{C}$. ∎