  We introduce the notion of Kan injectivity in 2-categories and study its
properties. For an adequate 2-category $\mathcal{K}$, we show that every set of
morphisms $\mathcal{H}$ induces a KZ-pseudomonad on $\mathcal{K}$ whose
2-category of pseudoalgebras is the locally full sub-2-category of all objects
(left) Kan injective with respect to $\mathcal{H}$ 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.
