@Michael

The puzzle asks us to cook up suitable functors \$$R_1, R_2, R_3\$$ and find left adjoints for them, and possibly right adjoints. Currently I can't even cook up the first two functors!

Re the natural isomorphism \$$\mathcal{D}[LX, Y] \cong \mathcal{C}[X, RY]\$$, I've found that you don't usually need to worry too much about naturality. The key thing is to look for a bijection between \$$\mathcal{D}\$$-arrows \$$LX \rightarrow Y\$$ and \$$\mathcal{C}\$$-arrows \$$X \rightarrow RY\$$. Usually something will "jump out" at you, and it will usually turn out to be natural.

What the naturality condition actually says is that if we write \$$\Phi\$$ for our bijection, we ought to have

\$\Phi(Lf \circ x \circ g) = f \circ \Phi(x) \circ Rg\$

for any arrows \$$f\$$ into \$$X\$$ and \$$g\$$ out of \$$Y\$$. But this tends to fall out automatically for any sensible choice of \$$\Phi\$$.