Global Kan extensions as adjoints to restriction

Let $K:\mathcal{C}\to \mathcal{D}$ be a functor with $\mathcal{C}$ and $\mathcal{D}$ small, and let $\mathcal{E}$ be locally small (Small, locally small, and large categories, If $\mathcal{C}$ is small and $\mathcal{D}$ is locally small then $[\mathcal{C},\mathcal{D}]$ is locally small; if both are small it is small). Then the functor categories $[\mathcal{C},\mathcal{E}]$ and $[\mathcal{D},\mathcal{E}]$ are legitimate categories (Functor category $[\mathcal{C},\mathcal{D}]$).

Precomposition with $K$ defines the restriction functor

$K^{\ast }:[\mathcal{D},\mathcal{E}]⟶[\mathcal{C},\mathcal{E}],H⟼HK\mathrm{.}$

A global left Kan extension along $K$ is a functor

${\mathrm{Lan}}_K:[\mathcal{C},\mathcal{E}]⟶[\mathcal{D},\mathcal{E}]$

equipped with an adjunction ${\mathrm{Lan}}_K⊣K^{\ast }$ in the sense of Adjunction by unit, counit, and the triangle identities.

Dually, a global right Kan extension along $K$ is a functor

${\mathrm{Ran}}_K:[\mathcal{C},\mathcal{E}]⟶[\mathcal{D},\mathcal{E}]$

equipped with an adjunction $K^{\ast }⊣{\mathrm{Ran}}_K$.

This is a functor-level notion. It differs from a local Kan extension of one functor $F:\mathcal{C}\to \mathcal{E}$ in Left and right Kan extensions: forming a global Kan extension requires data for every object of the functor category, not a silent class-indexed choice of one local extension for each $F$.