@John Lol I've been busy catching up. This stuff is great!

This first picture shows the relationship between all the functors and \$$\mathrm{Lan}_{G}(H)\$$ used in this Lecture.

![Left Kan Extension](http://aether.co.kr/images/left_kan_extension.svg)

I can guess where to draw \$$\mathrm{Ran}_{G}(H)\$$ but don't know the details yet so I have left it out.

So now onto **Puzzle 155-157**. I used the most obvious example which is send each person to the same person and drew them out in full detail.

![Puzzle 155-157](http://aether.co.kr/images/kan_extension_example.svg)

The picture shows \$$\beta:H \Rightarrow F \circ G \$$ on the left and \$$\alpha: \mathrm{Lan}\_{G}(H) \Rightarrow F \$$ on the right. The naturality square has become a naturality circle. You can follow using Keith's diagram in Comment 14. And as you can see \$$\beta_{\mathrm{Germans}} = \alpha_{\mathrm{Germans}}\$$.