TITLE: Toy examples for Kan extensions
QUESTION [29 upvotes]: Background: If $\mathcal{C}$ is a cocomplete category and $f : I \to J$ is a functor between small categories, then $f^* : \mathrm{Hom}(J,\mathcal{C}) \to \mathrm{Hom}(I,\mathcal{C})$ has a left adjoint $\mathrm{Lan}(f)$, the left Kan extension along $f$. One may express this as the following coend:
$$\mathrm{Lan}(f)(F) = \int^i \hom(f(i),-) \otimes F(i).$$
What are some toy examples for left Kan extensions? I know that left Kan extensions are generalizations of colimits, that they are useful for constructing pullback functors of presheaves and the definition of left derived functors in the context of model categories as well as homological algebra. But I would like to see some specific easy examples which are perhaps not really important, but show what's going on.
Here is an example: Consider the inclusion $f : \{0,1\} \hookrightarrow \{0<1\}$. The left Kan extension corresponds to the functor $\mathcal{C} \times \mathcal{C} \to \mathrm{Mor}(\mathcal{C})$ which maps a pair of objects $(A,B)$ to the morphism $(A \to A \oplus B)$.
I am not looking for well-known general classes of examples (geometric realization, tensor products, etc.).

REPLY [4 votes]: If you like simplicial sets: We have an adjunction

where

$\mathrm{inj}$ is the functor sending $\star$ to $[0]$ and $\mathrm{id}_{\star}$ to $\mathrm{id}_{[0]}$.
$\mathrm{pr}$ is the unique functor from $\Delta$ to $\mathsf{pt}$.

Taking left/right Kan extensions then gives you the usual quadruple adjunction between $\mathsf{Sets}$ and $\mathsf{sSets}$:

You also have for each $n\in\mathbb{N}$ a functor $\mathrm{inj}_n\colon\mathsf{pt}\longrightarrow\Delta$ sending $\star$ to $[n]$ and $\mathrm{id}_{\star}$ to $\mathrm{id}_{[n]}$. Taking Kan extensions gives you a triple adjunction, where one of the functors (the precomposition one) is $\mathrm{ev}_n$, the functor sending a simplicial set $X_\bullet$ to $X_n$.