ScalingStacks

Proof. Condition (R.1) implies both that $i_{*}$ carries $T$-local objects to $S$-local objects and that we obtain an adjunction:

$$L^{T}\circ i^{*}\colon S^{-1}\pre(\Upsilon_{n})\rightleftarrows T^{-1}\pre(\mathcal{R})\colon i_{*}.$$

Similarly, condition (R.2) implies that $i^{*}$ carries $S$-local objects to $T$-local objects and that we obtain a second adjunction:

$$L^{S}\circ i_{!}\colon T^{-1}\pre(\mathcal{R})\rightleftarrows S^{-1}\pre(\Upsilon_{n})\colon i^{*}.$$

Since $i^{*}$ sends $S$-local objects to $T$-local objects, $i^{*}\simeq L^{T}\circ i^{*}$ when restricted to the $S$-local objects of $\pre(\Upsilon_{n})$. Thus $i^{*}\colon\pre(\Upsilon_{n})\to\pre(\mathcal{R})$ restricts to a functor

$$i^{*}\colon\cat_{(\infty,n)}=S^{-1}\pre(\Upsilon_{n})\to T^{-1}\pre(\mathcal{R}),$$

that admits a left adjoint $L^{S}\circ i_{!}$ and a right adjoint $i_{*}$.

Notice that $i_{!}(R)\cong i(R)$ in $\pre(\Upsilon_{n})$, where we have identified $\mathcal{R}$ and $\Upsilon_{n}$ with their images under the Yoneda embedding in, respectively, $\pre(\mathcal{R})$ and $\pre(\Upsilon_{n})$. Thus by (R.3) the counit map applied to $r\in\mathcal{R}$,

$$R\to i^{*}\circ L^{S}\circ i_{!}(R)\cong i^{*}\circ L^{S}i(R)\cong i^{*}i(R)$$

becomes an equivalence in $T^{-1}\pre(\mathcal{R})$ (the last equality follows from Lemma 8.6, as the image of $i$ consists of $S$-local objects). The endofunctor $i^{*}\circ L^{S}\circ i_{!}\colon T^{-1}\pre(\mathcal{R})\to T^{-1}\pre(\mathcal{R})$ is a composite of left adjoints, hence commutes with colimits. Therefore, as $\mathcal{R}$ is dense in $T^{-1}\pre(\mathcal{R})$, the functor $i^{*}\circ L^{S}\circ i_{!}$ is determined by its restriction to $\mathcal{R}$. It is equivalent the left Kan extension of its restriction to $\mathcal{R}$. Consequently $i^{*}\circ L^{S}\circ i_{!}$ is equivalent to the identity functor.

On the other hand, for each $X\in\cat_{(\infty,n)}$, consider the other counit map $X\to i_{*}i^{*}X$. For each $k$, we have natural equivalences,

	$\displaystyle\map(C_{k},i_{*}i^{*}X)$	$\displaystyle\simeq\map(i(R_{k}),i_{*}i^{*}X)$
		$\displaystyle\simeq\map(i^{*}i(R_{k}),i^{*}X)$
		$\displaystyle\simeq\map(R_{k},i^{*}X)$
		$\displaystyle\simeq\map(i(R_{k}),X)\simeq\map(C_{k},X),$

which follow from (R.3), (R.4), the identity $i_{*}(R_{k})\cong i(R_{k})$, and the fact that $i^{*}X$ is $T$-local. By Remark 7.1 this implies that the counit $X\to i_{*}i^{*}X$ is an equivalence. Thus $i^{*}$ is a functor with both a left and right inverse, hence is itself an equivalence of $\infty$-categories. ∎