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.).
Best Answer
(I hope I didn't mess any of these up.) EDIT: I did mess a couple up, the seond and third, many thanks to @JoshuaMeyers for catching the mistakes! They have been corrected.
For $f : B\mathbb{N} \to B\mathbb{Z}$ (where $BM$ is the one-object category corresponding to the monoid $M$) given by the inclusion, $\mathrm{Lan}(f)$ sends an endomorphism $g:X \to X$ to the endomorphism of $Y := \mathrm{colim}(X \xrightarrow{g} X \xrightarrow{g} \cdots)$ that is given by $g$ on every copy of $X$. This is the usual construction for "inverting $g$".
Let $I = \{0 \to 1\}$. For $f : I \to B \mathbb{N}$, the morphism that picks out of the generator of $\mathbb{N}$, $\mathrm{Lan}(f)$ sends a morphism $g : X \to Y$ to the endomorphism on $X \sqcup \coprod_\mathbb{N} Y$ mapping $X$ to the first summand $Y$ by means of $g$, and sending each summand $Y$ to the next via the identity.
Let $I$ be as above and $J$ be the category with two objects and a unique isomorphism between them. For the inclusion $f : I \to J$, again $\mathrm{Lan}(f)$ sends a morphism $g: X \to Y$ to the identity on $Y$ (ignoring $X$ and $g$).
Let $P = I \coprod_{\{0,1\}} I$ be the pair of parallel arrows. For the fold map $f : P \to I$, $\mathrm{Lan}(f)$ sends a pair $g, h : X \to Y$ to the canonical morphism $X \to \mathrm{coeq}(g,h)$ (note the domain is $X$, not $Y$).
(Before you scold me for this one, consider that one person's "proposition" is another's "family of examples".) If $f : C \to D$ is an opfibration, then $\mathrm{Lan}(f)$ is computed by taking colimits over the fibres: it sends a functor $g : C \to E$ to $d \mapsto \mathrm{colim}(g|_{f^{-1}(d)})$, where $f^{-1}(d)$ is category consisting of the objects of $C$ mapping to $d \in D$ and those morphisms between them that map to the identity on $d$.