Theorem: Let $\mathcal{C}$ and $\mathcal{E}$ be two categories and let $F:\mathcal{C}\to \mathcal{E}$ be a functor. Let $\bullet $ be the terminal category consisting of a unique object $\bullet$ and a unique morphism. Then a colimit of $F$ is a left Kan extension of $F$ along $K:\mathcal{C}\to \bullet$, i.e. $\mathrm{Lan}_KF(\bullet)=\mathrm{colim} F$.