Graduated categories of presheaves

We study conditions on a small category $\mathcal{A}$ under which every finitely generated object of its presheaf category is finitely presentable. Example: this holds for a group $\mathcal{A}$ iff $\mathcal{A}$ is Noetherian. For ordinals $\mathcal{A}=α$ this holds iff $α\leq ω$, whereas this alway