The following hold.
Let \(\mathcal C\in \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) and \(S\) a small set of compact-projective generators. Then the set \(S\subseteq \mathcal C^{\mathrm{cp}}\) generates the full subcategory \(\mathcal C^{\mathrm{cp}}\) under retracts and finite coproducts. In particular, every compact-projective object in \(\mathcal C\) is a retract of a finite coproduct of objects in \(S\).
Let \(\mathcal C\in \mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) and \(S\) a small set of compact generators. Then the set \(S\subseteq \mathcal C^{c}\) generates the full subcategory \(\mathcal C^{c}\) under retracts and finite colimits. In particular, every compact object in \(\mathcal C\) is a retract of an iterated finite colimit of objects in \(S\).