A presentable abelian category \(\mathcal A\) is \(1\)-projectively generated if it is compactly generated and if the full subcategory of compact objects \(\mathcal A^{\mathrm{c}}\) has enough projective objects, i.e. if for every compact object \(a\in \mathcal A\) there exists a compact \(1\)-projective object \(p\) and an epimorphism \(p \twoheadrightarrow a\). In particular, this implies that also \(\mathcal A\) has enough projective objects, i.e. that for every object \(a\in \mathcal A\) there exists a \(1\)-projective \(p\) and an epimorphism \(p \twoheadrightarrow a\).
For example, the abelian category \(\mathrm{mod}_k\) is a \(1\)-projectively generated presentable \(1\)-category with \(\mathrm{mod}_k^{\mathrm{c}}\) the full subcategory of finitely generated modules and \(\mathrm{mod}_k^{\mathrm{c}1\mathrm{p}}\) the full subcategory of finitely generated projective \(k\)-modules.