ScalingStacks

Proof. By Axiom (C.1), the left Kan extension of $f$ along the Yoneda embedding is a localization $F\colon\pre(\gaunt_{n}^{\omega})\to\mathcal{C}$; the right adjoint is a fully faithful functor $G\colon\mathcal{C}\hookrightarrow\pre(\gaunt_{n}^{\omega})$. Write $W$ for the class of morphisms of $\pre(\gaunt_{n}^{\omega})$ that are carried to equivalences of $\mathcal{C}$ by $F$, so that $\mathcal{C}\simeq W^{-1}\pre(\gaunt_{n}^{\omega})$. The class $W$ is strongly saturated, and by [28, Proposition 5.5.4.16], it is of small generation.

By Axiom (C.4), the class $W$ contains the morphisms of Notation 6.5. We claim further that $T_{0}\subseteq W$. To prove this, it suffices to show that $W$ is stable under the operation $H\times_{C_{i}}(-)$ for any $H\in\gaunt_{n}^{\omega}$.

So let $W^{\prime}\subseteq W$ be the subset consisting of those morphisms $\phi\colon X\to Y$ of $W$ such that for any morphism $Y\to C_{i}$ and any morphism $H\to C_{i}$ of $\gaunt_{n}^{\omega}$, the pullback $H\times_{C_{i}}\phi\colon H\times_{C_{i}}X\to H\times_{C_{i}}Y$ also lies in $W$. By the universality of colimits in $\pre(\gaunt_{n}^{\omega})$, it follows that $W^{\prime}$ is closed under colimits. From Proposition 8.5 for $\mathcal{R}=\gaunt_{n}^{\omega}$, $i=\id$, and $U=W$, we deduce that since $(\mathcal{C},f)$ satisfies Axiom (C.3), there is a subset $W_{0}\subseteq W$ that generates $W$ under colimits and is stable under the operation $H\times_{C_{i}}(-)$ for any $H\in\gaunt_{n}^{\omega}$. Hence $W_{0}\subseteq W^{\prime}$, and so $W^{\prime}=W$.

Since $T_{0}\subseteq W$ (and thus $T\subseteq W$), it follows that $F$ factors through a left adjoint $\precat_{(\infty,n)}\to\mathcal{C}$, which by a small abuse we will also call $F$. Composing this left adjoint with the fully faithful left adjoint $j_{!}\colon\cat_{(\infty,n)}\hookrightarrow\precat_{(\infty,n)}$, we obtain our desired left adjoint $K\mathrel{\mathop{:}}=Fj_{!}$.

To construct the desired natural transformation $\eta$, compose the counit $j_{!}j^{*}\to\id$ with $F$ to obtain $Kj^{*}\to F$, and then restrict along Yoneda to obtain $\eta\colon Kg\to f$. By definition, $\eta$ is an equivalence when restricted to $\Upsilon_{n}$, and thus a fortiori when restricted to $\mathbb{G}_{n}$. ∎