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 \( \mathcal A =\alpha \) this holds iff \( \alpha\le \omega \), whereas this always holds for \( \mathcal A = \alpha^{op} \). Various important properties of set functors also apply to endofunctors on locally finitely presentable categories which are graduated. This means that every finitely presentable object \( X \) carries a grade (in \( \mathbb N \)) and grades respect subobjects and strong quotients of \( X \). We characterize presheaf categories \( \mathbf{Set}^{\alpha^{op}} \) which are graduated for three types of small categories \( \mathcal A \). If \( \mathcal A \) is a poset, all down sets of elements must be finite. For a group \( \mathcal A \), a finite bound on the length of a chain of subgroups must exist. In the case of cartesian categories \( \mathcal A \), every object must carry a finite number of sieves. Example: all finitary set functors form a graduated category, here \( \mathcal A \) is the dual of finite sets. A closely related concept is a locally finitely presentable category with the descending chain condition (a DCC category): every finitely presentable object has only finite descending chains of subobjects or strong quotients. The presheaf category on a group \( G \) is DCC iff \( G \) has no infinite chain of subgroups. Example: presheaves on the group \( \mathbb Z \) are not DCC, but finite generation implies finite presentation.