I'm currently going through the proof of the $\mathbf {Special ~ adjoint ~ functor ~ theorem}$ (SAFT) in Saunders Mac Lane's "Categories for the Working Mathematician" and I'm having trouble with the following line in the proof:
"Conversely, it suffices as usual to construct for each $x\in X$ an initial object in the comma category $D = (x \downarrow G)$"
If this does indeed suffice, then I'm OK with the rest of the proof. I'm just not sure why "having for each $x\in X$ an initial object in $D = (x \downarrow G)$" suffices to construct a left adjoint for $G$ (because after all the goal is to construct a left adjoint for $G$).
This is the first time I've had to deal with comma categories (or slice categories) so I'm not too familiar with their properties. I've tried searching on the internet for some standard results about initial objects in slice categories with not much luck.
If someone can please provide a reference which justifies this claim or explain/provide a proof themselves that would be extremely helpful and very much appreciated!
Thanks!
This follows from Theorem IV.1.2(ii) (page 83) in Mac Lane (a "universal arrow from $x$ to $G$" is just a different name for an initial object of $(x\downarrow G)$). The idea is that if $x\to Gy$ is an initial object of the category $(x\downarrow G)$, then you can define $Fx=y$. If such an initial object exists for each $x$, this operation $F$ can be turned into a functor that is a left adjoint for $G$.