Kan Extension

Definition

Let $F:\mathcal{C}\Rightarrow \mathcal{E};K:\mathcal{C}\to \mathcal{D}$ be functors. A left Kan extension of $F$ along $K$ is a functor ${\mathrm{Lan}}_{K}F:\mathcal{D}\Rightarrow \mathcal{E}$ Together with a natural transformation $\eta :F\Rrightarrow {\mathrm{Lan}}_{K}F$ such that any other pair $(G:\mathcal{D}\Rightarrow \mathcal{E},\epsilon :F\Rrightarrow G)$