Left and right Kan extensions

Let $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$ be categories, let $K:\mathcal{C}\to \mathcal{D}$ be a functor, and let $F:\mathcal{C}\to \mathcal{E}$ be a functor (Covariant functor, identity functor, composite functor, and contravariant functor).

A left Kan extension of $F$ along $K$ is a functor $L:\mathcal{D}\to \mathcal{E}$ together with a natural transformation

$\eta :F\Rightarrow LK$

(Natural transformation and its components) such that for every functor $M:\mathcal{D}\to \mathcal{E}$ and every natural transformation $\alpha :F\Rightarrow MK$, there exists a unique natural transformation $\bar{\alpha }:L\Rightarrow M$ with

$\alpha =(\bar{\alpha }K)\circ \eta \mathrm{.}$

Thus $(L,\eta )$ is initial among pairs $(M,\alpha )$ with $\alpha :F\Rightarrow MK$.

A right Kan extension of $F$ along $K$ is a functor $R:\mathcal{D}\to \mathcal{E}$ together with a natural transformation

$\epsilon :RK\Rightarrow F$

such that for every functor $M:\mathcal{D}\to \mathcal{E}$ and every natural transformation $\beta :MK\Rightarrow F$, there exists a unique natural transformation $\bar{\beta }:M\Rightarrow R$ with

$\beta =\epsilon \circ (\bar{\beta }K)\mathrm{.}$

Thus $(R,\epsilon )$ is terminal among pairs $(M,\beta )$ with $\beta :MK\Rightarrow F$.