Let \(U\) be a universe.

• -Let \(\kappa \) be a \(U\)-small sound doctrine. We say a category \(C\) is \((U,\kappa )\)-presentable if there exist a \(U\)-small category \(A\), a reflective full subcategory \(L\) of the category of \(U\)-small presheaves on \(A\) closed under \(U\)-small \(\kappa \)-filtered colimits, and an equivalence \(C\simeq L\).
• -We say a category \(C\) is \(U\)-presentable if it is \((U,\kappa )\)-presentable for some \(U\)-small sound doctrine \(\kappa \).