Semantic Factorization and Descent

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 , either one existing if other does. The result can seen counterpart account celebrated Bénabou–Roubaud theorem. This leads in particular monadicity theorem, since it characterizes via descent. It should noted all conditions on trivially hold whenever has left adjoint and, hence, this case, we find exact condition namely, an effective faithful .