Puzzle 157: Using Christopher's answer to 156 we should get the natural transformation which sends every German to Heinrich. The idea is that a natural transformation \\(\alpha : \mathrm{Lan}_G (H) \rightarrow F \\) is uniquely determined by where it sends the Germans because naturality will always give a unique map for the Italians. This map which \\(\alpha\\) is uniquely determined by should be the map \\(\beta\\) because there is a bijection between such maps from the adjunction equations.

Puzzle 158: I'll start by answering the second part because it is easier. Because there are no naturality conditions to check, this number should be the number of functions between the Germans in the image of \\(F\\) to the Germans in the image of \\(H\\). That number should be \\(|H(\mathrm{Germans})|^{|F(\mathrm{Germans})|} = 5^4\\). To find the number of natural transformations \\(\alpha : \mathrm{Lan}_G (H) \rightarrow F\\) we note that by naturality of \\(\alpha\\), \\(\alpha\\) is uniquely determined by its function from the germans in \\(\mathrm{Lan}_G (H)\\) to the Germans in \\(F\\). So this number should also be \\(5^4\\).

Puzzle 159: Is it the arrow category and the terminal category respectively?