Limits and colimits are Kan extensions along the functor to the terminal category

Let $\ast$ be the unique object of $\mathbf{1}$. An object of $(!\downarrow \ast )$ is an object $c$ of $\mathcal{C}$ together with the unique arrow $!c\to \ast$, and a morphism is exactly a morphism of $\mathcal{C}$; so $(!\downarrow \ast )$ is canonically just $\mathcal{C}$. The same argument shows $(\ast \downarrow !)$ is also canonically $\mathcal{C}$.

Under the identification of step 1.1, the comma-category colimit formula of [L1] says that a left Kan extension value at $\ast$ is an initial cocone under the original diagram $F$. By [F1], that is exactly a colimit of $F$.

Under the same identification, the comma-category limit formula of [L1] says that a right Kan extension value at $\ast$ is a terminal cone over $F$. By [F1], that is exactly a limit of $F$.