ScalingStacks

Proof.

We prove the first statement, the second is analogous. The \(\infty\)-category \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) may be understood as the subcategory of \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^\mathrm{L})\) on those presentable \(\mathrm{Sp}_{\geq 0}\)-module \(\infty\)-categories \(\mathcal C\) whose underlying \(\infty\)-category is projectively generated and for which the action functor \(\mathrm{Sp}_{\geq 0}\otimes \mathcal C\rightarrow\mathcal C\) preserves compact-projectives, and those cocontinuous \(\mathrm{Sp}_{\geq 0}\)-module functors \(\mathcal C\rightarrow\mathcal D\) for which the underlying functor preserves compact projectives. In particular, the equivalence \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^\mathrm{L}) \rightarrow {\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) restricts to a fully faithful functor \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) \rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\). It therefore suffices to verify that for an additive presentable \(\infty\)-category \(\mathcal C\), the action \(\mathrm{Sp}_{\geq 0} \times \mathcal C\rightarrow\mathcal C\) sends a pair of compact-projective objects \((a, b) \in \mathrm{Sp}_{\geq 0}^{\mathrm{cp}} \times \mathcal C^{\mathrm{cp}}\) to a compact-projective of \(\mathcal C\). This follows since any compact projective in \(\mathrm{Sp}_{\geq 0}\) is generated under finite coproducts and retracts by the unit object \(\mathbb{S}\); see lemma 3.2.9. ◻