Michael - very nice pictures! I especially like this one:



At first I thought the blue arrow should point from \\(\alpha\\) to \\(\beta\\), because the definition of left Kan extension says there's a one-to-one and onto function from the set of

\[ \text{ natural transformations } \alpha: \text{Lan}_G(H) \Rightarrow F \]

to the set of

\[ \text{ natural transformation } \beta: H \Rightarrow F \circ G \]

Since it's one-to-one and onto, we could have a double-headed arrow pointing from \\(\alpha\\) to \\(\beta\\) and also back from \\(\beta\\) to \\(\alpha\\).

But then I realized your blue arrow pointing from \\(H\\) to \\(\text{Lan}\_G (H) \\) is also a real thing, it's the _process of left Kan extending along \\(G\\)_, namely the functor

\[ \text{Lan}\_G : \mathbf{Set}^\mathcal{D} \to \mathbf{Set}^\mathcal{C} .\]

Whew, there are lots of arrows here!