ScalingStacks

Proof. By the previous proposition, $G$ is a local object, and the cofiber

$$\Sigma i_{!}i^{*}R^{\lor\infty}\simeq i_{!}\Sigma i^{*}R^{\lor\infty}$$

of $R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}G$ is in the image of $i_{!}$. $G$ is compact because $R^{\lor\infty}$ is a compact generator of $F^{\infty}_{R}$ and the functor $\Psi(F^{\infty}_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ preserves compact objects [70, 2.9]. ∎