Let \(\mathcal C\) be a cocomplete \(\infty\)-category.
The full subcategory \(\mathcal C^{\mathrm{c}}\) of compact objects is closed under retracts and finite colimits [Lur09, Cor. 5.3.4.15 and Rem. 5.3.4.16] and hence yields an object \(\mathcal C^\mathrm{c}\in \mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\). This defines a functor \((-)^{\mathrm{c}} \colon \mathrm{Pr}^{\mathrm{L},\mathrm{c}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\).
The full subcategory \(\mathcal C^{\mathrm{cp}}\) of compact-projective objects is closed under retracts and finite coproducts [Lur09, Rem. 5.5.8.19] and hence defines an object \(\mathcal C^{\mathrm{cp}} \in \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\). This defines a functor \((-)^{\mathrm{cp}} \colon \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\).