[0041]
Proof.
The first statement is [Lur17, Lem. 5.3.2.9] for \(\kappa=\omega\), also see [Lur09, Prop. 5.5.7.8].
To prove the second statement, we note that [Lur09, Cor. 5.3.6.10, Rem. 5.5.8.16] implies that \(\mathcal P^{\Sigma}(-)\) defines a functor from \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) to the huge \(\infty\)-category \(\widehat{\mathrm{Cat}}_{\infty}^{\mathrm{cocpl}}\) of large \(\infty\)-categories which admit all small colimits and colimit preserving functors. By [Lur09, Prop. 5.5.8.10], \(\mathcal P^{\Sigma}(\mathcal C)\) is an accessible localization of \(\mathcal P(\mathcal C)\) and hence is presentable, so that \(\mathcal P^{\Sigma}\) factors through the full subcategory \(\mathrm{Pr}^\mathrm{L}\) of \(\widehat{\mathrm{Cat}}_{\infty}^{\mathrm{cocpl}}\). By [Lur09, Prop. 5.5.8.22], every object in the image of the Yoneda embedding \(\mathcal C\hookrightarrow
\mathcal P^{\Sigma}(\mathcal C)\) is compact-projective, and since \(\mathcal P^{\Sigma}(\mathcal C)\) is a localization of \(\mathcal P(\mathcal C)\), it is generated under small colimits by objects in \(\mathcal C\); hence \(\mathcal P^{\Sigma}(\mathcal C)\) is projectively generated. Moreover, by [Lur09, Prop. 5.5.8.25], the compact-projective objects of \(\mathcal P^{\Sigma}(\mathcal C)\) are precisely the objects in (the essential image of) \(\mathcal C\) (note: this uses that \(\mathcal C\) is idempotent complete). Therefore, for any morphism \(f\colon \mathcal C\rightarrow\mathcal D\) in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) the cocontinuous functor \(\mathcal P^{\Sigma}(f)\colon
\mathcal P^{\Sigma}(\mathcal C) \rightarrow\mathcal P^{\Sigma}(\mathcal D)\) preserves compact-projectives; hence \(\mathcal P^{\Sigma}\colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Pr}^\mathrm{L}\) factors through the subcategory \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow
\mathrm{Pr}^\mathrm{L}\). To show that it factors as an equivalence, notice that it is fully faithful since for \(\mathcal C, \mathcal D\in \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), \[\mathrm{Fun}^{\sqcup}(\mathcal C, \mathcal D) \simeq
\mathrm{Fun}^{\sqcup}(\mathcal C, \mathcal P^{\Sigma}(\mathcal D)^{\mathrm{cp}}) \simeq\mathrm{\mathrm{Fun}^{L, cp}} (\mathcal P^{\Sigma}(\mathcal C),
\mathcal P^{\Sigma}(\mathcal D))\] where the first equivalence uses that \(\mathcal D\simeq
\mathcal P^{\Sigma}(\mathcal D)^{\mathrm{cp}}\) and the second equivalence uses that for any presentable \(\mathcal E\), the map \(\mathrm{Fun^L}(\mathcal P^{\Sigma}(\mathcal C), \mathcal E) \rightarrow\mathrm{Fun}^{\sqcup}(\mathcal C, \mathcal E)\) is an isomorphism (this follows e.g. from [Lur09, Prop. 5.5.8.10]) and the fact that \(\mathcal C\simeq \mathcal P^{\Sigma}(\mathcal C)^{\mathrm{cp}}\). Lastly, surjectivity on objects follows since by [Lur09, Prop. 5.5.8.25] any projectively generated presentable \(\infty\)-category \(\mathcal D\) is equivalent to \(\mathcal P^{\Sigma}(\mathcal C)\) where \(\mathcal C\) is the smallest full subcategory of \(\mathcal D\) spanned by finite coproducts of objects in the set \(S\) of compact-projective generators.
The third statement follows since the induced composite \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) sends \(\mathcal C\) to \(\mathcal P^{\Sigma}(\mathcal C)^{\mathrm{c}}\), which in the notation of proposition 3.1.11 is equivalent to \(\mathcal P_{\mathcal K}^{\mathcal K'}(\mathcal C)\) for \(\mathcal K\) the collection of finite sets together with the ‘walking idempotent’ \(\mathrm{Idem}\) of [Lur09, Sec. 4.4.5] and \(\mathcal K'\) the collection of finite categories together with \(\mathrm{Idem}\). Hence, by proposition 3.1.11.([003D]) this functor \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) is left adjoint to the forgetful functor \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\rightarrow
\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\). The fourth statement then follows from the previous ones and proposition 3.1.11.([003C]), since \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) and \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) are presentable and since the functor \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) is equivalent to \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) and hence a left adjoint. ◻