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!