KZ-Pseudomonads and Kan Injectivity

We introduce the notion of Kan injectivity in 2-categories and study its properties. For an adequate 2-category K, we show that every set of morphisms H induces a KZ-pseudomonad on K whose 2-category of pseudoalgebras is the locally full sub-2-category of all objects (left) Kan injective with respect to H and morphisms preserving Kan extensions. The main ingredient is the construction of a (pseudo)chain whose appropriate “convergence” is ensured by a small object argument.