I'll take a stab at puzzle 157.

> **Puzzle 157.** Which natural transformation \\(\beta: H \Rightarrow F \circ G\\) corresponds to the natural transformation \\(\alpha :\textrm{Lan}\_G(H) \Rightarrow F\\) you gave in Puzzle 155?

I think that given a natural transformation \\(\alpha\\), we can obtain a natural transformation \\(\beta\\) by inheriting the "Germans" component, that is \\(\beta_{\mathrm{Germans}} = \alpha_{\mathrm{Germans}}\\).
For Christopher's example, from [the first comment](https://forum.azimuthproject.org/discussion/comment/19649/#Comment_19649), the natural transformation \\(\beta\\) would be:

\beta_{\mathrm{Germans}} = \left\\{ \mathrm{Ilsa} \mapsto \mathrm{Heinrich}, \mathrm{Klaus} \mapsto \mathrm{Heinrich}, \mathrm{Jorg} \mapsto \mathrm{Heinrich}, \mathrm{Sabine} \mapsto \mathrm{Heinrich} \right\\}

However, I'm not sure how I can obtain this sort of correspondence for the general case:
given a pair of adjoint functors \\(F \dashv G\\), I understand that there exists a one-to-one correspondence between \\(\mathcal{D}(F(A), B)\\) and \\(\mathcal{C}(A, G(B))\\),
but I don't know how to get a hold on the bijection \\(\phi : \mathcal{D}(F(A), B) \to \mathcal{C}(A, G(B))\\).