There's an exercise in the Handbook of Categorical Algebra (exercise 6.7.11, volume 1) where you're supposed to show the that the following theorem:
Given three categories $\mathcal{A}, \mathcal{B}, \mathcal{C}$ with $\mathcal{A}$ small and $\mathcal{C}$ cocomplete and two functors $G:\mathcal{A} \rightarrow \mathcal{C}$ and $F:\mathcal{A}\rightarrow\mathcal{B}$, the left Kan extension of $G$ along $F$ always exists.
is a corollary of the more general adjoint functor theorem:
Consider a functor $F:\mathcal{A}\rightarrow\mathcal{B}$ with $\mathcal{A}$ Cauchy complete. The following conditions are equivalent
- $F$ has a left adjoint
- $F$ is absolutely flat and satisfies the solution set condition for every object $B$ of $\mathcal{B}$
(but not of the general adjoint functor theorem, which you will recall is the one involving preservation of limits and requires that $\mathcal{A}$ be complete)
Now, as far as I can tell, the method of proof intended is to show that the functor $F^*:\mathcal{C}^\mathcal{B}\rightarrow\mathcal{C}^\mathcal{A}$ of precomposition with $F$ satisfies the conditions for the MGAFT, since the existence of Kan extensions along $F$ for every $G$ is equivalently the existence of reflections along $F^*$ for every object $G$ of $\mathcal{C}^\mathcal{A}$.
It's easy to see that $\mathcal{C}^\mathcal{B}$ is Cauchy complete, since it is cocomplete, but I run into trouble when trying to show that $F^*$ is absolutely flat.
Like, for every functor $G \in \mathcal{C}^\mathcal{A}$ I can show that the category of elements of $\text{Nat}(G,F^*-)$ is non-empty (the colimit cocone $p:G→\Delta_L$ which must exist since $\mathcal{A}$ is small and $\mathcal{C}$ is cocomplete gives us an object $(\Delta_L,p)$) but I can't seem to find any way to show any of the other cofiltration conditions...
What am I not seeing here?
