If $\mathcal{E}$ is cocomplete and $\mathcal{C}$ is small, then left Kan extension $\text{Lan}_F K$ exists for all $K: \mathcal{C} \to \mathcal{E}$ and any $F: \mathcal{C} \to \mathcal{D}$.

Dually: if $\mathcal{E}$ is complete, right Kan extensions exist.

Left Kan extension exists and is pointwise if for each $d \in \mathcal{D}$, the comma category $(F \downarrow d)$ is small and colimit $\text{colim}_{(c,f) \in (F \downarrow d)} K(c)$ exists.

Then: $(\text{Lan}_F K)(d) = \text{colim}_{F(c) \to d} K(c)$

This provides explicit computation via weighted colimits.

$F: \mathcal{C} \to \mathcal{D}$ has right adjoint if and only if $\text{Ran}_F \text{id}_\mathcal{C}$ exists. The right adjoint is $U = \text{Ran}_F \text{id}_\mathcal{C}$.

This characterizes adjunctions via Kan extensions, showing adjunctions are "representable" Kan extensions.

For functor category $[\mathcal{C}, \mathcal{E}]$ and $F: \mathcal{C} \to \mathcal{D}$: $\text{Lan}_F: [\mathcal{C}, \mathcal{E}] \to [\mathcal{D}, \mathcal{E}]$ is left adjoint to restriction $F^*: [\mathcal{D}, \mathcal{E}] \to [\mathcal{C}, \mathcal{E}]$.

This gives $\text{Lan}_F \dashv F^*$ adjunction.

For dense functor $F: \mathcal{C} \to \mathcal{D}$, the functor $\text{Lan}_F \circ F^*: [\mathcal{D}, \textbf{Set}] \to [\mathcal{D}, \textbf{Set}]$ is comonad, and presheaves are coalgebras.

Comparison functor in monadicity theorem can be expressed via Kan extensions. Functor $U$ is monadic iff certain Kan extension is an equivalence.