ScalingStacks

Proof. Suppose $G$ is not conservative. Then to prove the lemma, it suffices to exhibit a presentable $\infty$-category $\mathcal{D}$ also cotensored over $\mathcal{C}$ and a nontrivial right adjoint $j:\mathcal{D}\rightarrow\mathcal{M}^{\prime}$ cotensored over $\mathcal{C}$ such that $j\circ G$ is trivial.

Define $\mathcal{D}$ to be the full $\infty$-subcategory of $\mathcal{M}^{\prime}$ of $G$-acyclic objects, that is, objects $m\in\mathcal{M}$ such that $G(m)$ is trivial. Our first task is to show that $\mathcal{D}$ is indeed presentable.

Observe that $\mathcal{D}$ is equivalent to the fiber product $\mathcal{D}\simeq 0\times_{\mathcal{M}}\mathcal{M}^{\prime}$, where the limit is computed in the $\infty$-category $\rm Cat_{\infty}$ of $\infty$-categories. Recall by [L2, Proposition 5.5.3.13], the natural functor $\mathcal{P}r^{\rm L}\rightarrow{\rm Cat}_{\infty}$ preserves limits. Furthermore, the forgetful functor $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})\to\mathcal{P}r^{\rm L}$ also preserves limits since it has a left adjoint (given by induction).

Since the functor $G$ preserves colimits and is $\mathcal{C}$-linear, we may regard it as a morphism in $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})$. Thus $\mathcal{D}$ can be computed as a limit in $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})$, and so can be regarded as an object of $\mathcal{P}r^{\rm L}$. In other words, $\mathcal{D}$ is presentable and furthermore tensored over $\mathcal{C}$. Finally, since $\mathcal{D}$ is tensored over $\mathcal{C}$, it is automatically cotensored as well.

Now it remains to show that the inclusion $j:\mathcal{D}\rightarrow\mathcal{M}^{\prime}$ is indeed a right adjoint and cotensored over $\mathcal{C}$. Since $j$ preserves all limits and colimits (and in particular $\kappa$-filtered colimits), the adjoint functor theorem applies. Finally, since $G$ is cotensored over $\mathcal{C}$, $j$ is as well. ∎