Kan extension

Remarks

The usual notation is $\mathrm{Lan}_K F$ for the left and $\mathrm{Ran}_K F$ for the right Kan extension. Both are unique up to a unique natural isomorphism when they exist. Suppose these extensions exist for all $F$. Then, between the functor categories, $\mathrm{Lan}_K$ is left adjoint and $\mathrm{Ran}_K$ is right adjoint to the restriction $G \mapsto G \circ K$ from $[\mathcal{B}, \mathcal{C}]$ to $[\mathcal{A}, \mathcal{C}]$.

Example: let $\mathcal{B} = \mathbf{1}$ be the category with one object and one morphism, and $K$ the unique functor. A functor $G \colon \mathbf{1} \to \mathcal{C}$ is an object $c$, and $G \circ K$ is the constant functor at $c$. A natural transformation $F \Rightarrow G \circ K$ is then a cocone from $F$ to $c$. So the clause of the definition says exactly that $\eta$ is a universal cocone, and $\mathrm{Lan}_K F$ is the colimit of $F$.

Non-example: let $\mathcal{A}$ be the empty category, $\mathcal{B} = \mathbf{1}$ and $\mathcal{C}$ the category of sets. Then $F$ and $\eta$ are empty, and a left Kan extension is an initial object of $\mathcal{C}$, namely the empty set. The constant functor $L$ at a one-point set is not a left Kan extension. For $G$ the constant functor at the empty set there is no natural transformation $L \Rightarrow G$, so the existence part of the clause fails.