I was trying to read Kashiwara-Schapira proof of existence of Kan extension. When target category is complete, and the other categories are small.
My interpretation of Kan extensions and the idea for approaching the existence is as follows;
Kan extensions are approximate extension of a functor, approximation in the categorical sense will be in terms of morphisms, since we are in the functor category, it will be natural transformations. What we want is a limit in the category of all “right extensions”. i.e., any other extension has to factor through this “limit” extension. This is just saying it’s the best approximation from the left.
The theorem due to Kan states that, if $\mathcal{A}$ and $\mathcal{B}$ are small, and $\mathcal{C}$ is complete the right Kan extension exists.
This makes sense because functor category to $\mathcal{C}$ will also be complete and limits will exist and if $\mathcal{B}$ is locally small, then the functor category $\mathcal{C}^\mathcal{B}$ will be locally small.
We just have to make sure it’s an inductive system now. So, we have to make sure the collection of all right extensions is at most a set, and get a functor from that set to the extensions.
I expect this is what’s happening in the proof, but can’t parse through the proof. I did not like the proofs in other books because they invloved new definitions like end, coend or comma categories.
Best Answer
The definition of Kan extension as already asserts that it arises as a certain limit or colimit (by virtue of being the value of a left or right adjoint) of functors. The exsitence theorem however, does not amount to verifying directly that those limits or colimtis exist. Instead, the existence theorem asserts that the values of the extension may arise as limits or colimits of objects, and that if those limits or colimits exist, then they assemble into the extension.
Consider functors $\phi\colon J\to I$ and $\beta\colon J\to C$, and a right extension $\psi\colon I\to C$ whose counit natural trasformation $\psi\circ\phi\Rightarrow\beta$ with components $\psi\circ\phi(j)\to\beta(j)$. This extension determines for each morphism $i\to\phi(j)$ in $I$, a morphism $\psi(i)\to\psi\circ\phi(j)\to\beta(j)$.
The key idea is that these morphisms $\psi(i)\to\phi(j)$ associated to each $i\to\phi(j)$ can be interpreted as a family indexed by the category $I$ of cones with vertices $\psi(i)$ over diagrams in $C$ indexed by categories $J^i$, that is natural in the variable $i$.
Explicitly, following the notation of KS (Definition 1.2.16), for each object $i$ of $I$, an object of the category $J^i$ (a special kind of comma category) is a pairs of an object $j$ of $J$ and a morphisms $i\to\phi(j)$ in $I$. A morphism in $J^i$ from such an object $i\to\phi(j)$ to another object $i\to\phi(j')$ is a morphism $j\to j'$ in $J$ for which $i\to\phi(j')$ factors as $i\to\phi(j)\to\phi(j')$.
The diagram in $C$ indexed by $J^i$ takes a morphism between $i\to\phi(j)$ and $i\to\phi(j')$ given by $j\to j'$ in $J$ to the morphism $\beta(j)\to\beta(j')$. That sending an object $i\to\phi(j)$ to $\psi(i)\to\psi\circ\phi(j)\to\beta(j)$ is a cone over the diagram follows from naturality of the transformation $\psi\circ\phi\Rightarrow\beta$.
Naturality of the family of cones amounts to the fact that, given a morphism $i'\to i$ in $I$, pre-composition with $\psi(i')\to\psi(i)$ sends the cone with vertex $\psi(i)$ over the diagram indexed by $J^i$ to the cone with vertex $\psi(i')$ over the diagram indexed by $J^{i'}$.
Finally, given merely such a family of cones sending morphisms $i\to\phi(j)$ in $I$ to morphisms $\psi(i)\to\beta(j)$, evidently the morphisms $\psi(i')\to\psi(i)$ form a functor, and the morphisms $\psi(\phi(j))\to\beta(j)$ given by taking $i=\phi(j)$, form a natural transformation $\psi\circ\phi\Rightarrow\beta$.
Thus there is a bijective correspondence between right extensions and such natural families of cones over the diagrams indexed by $J^i$. Moreover, one can check that a factorization of one extension through another corresponds to a factorization of one natural family of cones through another natural family of cones. The existence of a right Kan extension amounts to having a terminal object in this category of natural families of cones. The existence theorem for right Kan extension asserts that such a terminal object exists if (but not only if!) each diagram in the family has a limit. Indeed, then their limiting cones assemble into a natural family of cones (using the universal property of limits), hence determine a right extension. Moreover, every other natural family of cones factors uniquely through that natural family of cones (again using the universal property of limits).