Abstract Let $${\mathbb {A}}$$ A be a 2-category with suitable opcomma objects and pushouts. We give direct proof that, provided that the codensity monad of morphism p exists is preserved by morphism, factorization given lax descent object two-dimensional cokernel diagram up to isomorphism same as semantic , ...

We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of categories and even marked filtrations, where one descends not only admissible subcategories but also preferred objects.