Let $\mathcal{C}$ and $\mathcal{D}$ be two preorders, $L: \mathcal{C} \to \mathcal{D}$, $R: \mathcal{D} \to \mathcal{C}$ a pair of functors such that for all objects $c \in \mathcal{C}$ and $d \in \mathcal{D}$ there are bijections $$ \theta_{c,d}: \text{Hom}_{\mathcal{D}}(Lc,d) \to \text{Hom}_{\mathcal{C}}(c,Rd). $$ Prove that the bijections are natural in $c$ and $d$ and hence $L$ and $R$ form an adjunction.
Hint: Suppose $\mathcal{A}$ is a category, $F,G: \mathcal{A} \to \mathbf{Set}$ is a pair of functors taking values in the full subcategory $\mathcal{B} \subset \mathbf{Set}$ whose objects are sets with one element and the empty set. Note that $\mathcal{B}$ is a preorder. Then show that any family of bijiections $\{\tau_a: F(a) \to G(a)\}_{a \in \mathcal{A}}$ form a natural isomorphism.
I'm very confused about the hint, so I don't even know where to start. The way I was taught to show the above bijections are natural in $c$ and $d$ is to show two corresponding diagrams commute:
$$\require{AMScd}
\begin{CD}
\text{Hom}_{\mathcal{D}}(Lc,d) @>{\theta_{c,d}}>> \text{Hom}_{\mathcal{C}}(c,Rd)\\
@V{(Lf)^*}VV @V{f^*}VV \\
\text{Hom}_{\mathcal{D}}(Lc^\prime,d) @>{\theta_{c^\prime,d}}>> \text{Hom}_{\mathcal{C}}(c^\prime,Rd)
\end{CD}$$
and
$$\require{AMScd}
\begin{CD}
\text{Hom}_{\mathcal{D}}(Lc,d) @>{\theta_{c,d}}>> \text{Hom}_{\mathcal{C}}(c,Rd)\\
@V{g_*}VV @V{(Rg)_*}VV \\
\text{Hom}_{\mathcal{D}}(Lc,d^\prime) @>{\theta_{c,d^\prime}}>> \text{Hom}_{\mathcal{C}}(c,Rd^\prime)
\end{CD}$$
for any $f:c' \to c$ in $\mathcal{C}$ and $g:d \to d'$ in $\mathcal{D}$.
Here why do we need to consider two arbitrary functors $F$ and $G$ and show that there is a natural isomorphism between them? How can we use this result to prove the existence of an adjunction?
The hint states a more general fact.
Once you prove the hint, the exercise follows easily: The formulas $$\forall c \in \mathcal C,\, \forall d \in \mathcal D : \quad \begin{split} F(c,d) = \operatorname{Hom}_{\mathcal{D}}(Lc,d) \\ G(c,d) = \operatorname{Hom}_{\mathcal{C}}(c,Rd) \end{split}$$ define a pair functors $F,G \colon \mathcal C \times \mathcal D^{\rm op} \to \mathsf{Set}$ that take values in $\mathcal B$ (because $\mathcal C$ and $\mathcal D$ are preorders), so that the family $$\{\theta_{c,d} \colon F(c,d) \to G(c,d)\}_{(c,d) \in \mathcal C \times \mathcal D^{\rm op}}$$ are the components of a natural isomorphism.