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))\$$.