When Kan extensions exist, they are unique up to unique natural isomorphism. This follows from universal property.

Kan extensions compose: $\text{Lan}_G (\text{Lan}_F K) \cong \text{Lan}_{G \circ F} K$ when both sides exist. Similarly for right Kan extensions.

Right adjoints preserve right Kan extensions; left adjoints preserve left Kan extensions. More precisely, if $F \dashv U$, then: $F \circ \text{Lan}_G K \cong \text{Lan}_G (F \circ K)$

When iterated Kan extensions exist: $\text{Lan}_F (\text{Lan}_G K) \cong \text{Lan}_{F \circ \pi_1} K$

where $\pi_1$ is appropriate projection. This is "Fubini theorem" for Kan extensions.

Functor $F: \mathcal{C} \to \mathcal{D}$ is dense if $\text{Lan}_F (\mathcal{Y}_\mathcal{C}) \cong \mathcal{Y}_\mathcal{D}$ where $\mathcal{Y}$ is Yoneda embedding. Dense functors generate target category.

Example: Inclusion of finite sets into all sets is dense.

For functor $F: \mathcal{C} \to \mathcal{D}$, the codensity monad $T = \text{Ran}_F F$ (when exists) is a monad. This generalizes construction of monads from adjunctions.