Let $(P,\leq{}_P)$, $(Q,\leq{}_Q)$ partially ordered sets and $\alpha:P\longrightarrow{Q}$, $\beta:Q\longrightarrow{P}$ monotone functions satisfying the following property: given $p\in P$, $q\in Q$, then $\beta (q)\leq{}_P \hspace{0.1cm} p$ if and only if $q\leq{}_Q \hspace{0.1cm} \alpha (p)$.
We define the functors $\mathcal{F}:C_P\longrightarrow{C_Q}$, $\mathcal{G}:C_Q\longrightarrow{C_P}$ in the way that for each $p\in C_P$ we have that $\mathcal{F}p:=\alpha (p)$ and for each $q\in C_Q$ we have that $\mathcal{G}q:=\beta (q)$; where $C_P, C_Q$ are categories defined by the partially ordered sets $(P,\leq{}_P)$, $(Q,\leq{}_Q)$ respectively.
Assertion: $Hom_{C_Q}(\mathcal{F}-,-)\overset{\tau}{\cong}{Hom_{C_P}(-,\mathcal{G}-)}$. Indeed; for each $(p,q)\in C_P^{op}\times{C_Q}$, I have to define the map $\tau_{(p,q)}:Hom_{C_Q}( \mathcal{F}p,q)\longrightarrow{Hom_{C_P}(p,\mathcal{G}q)}$. My question is how to define this last application to then see that $\tau$ is a natural transformation and then the isomorphism.
Thank you very much.
For example, the naturality in the first variable amounts to the commutativity of the following diagram, for every $q\in Q$ and $p\le p'$ in $P$. $\require{AMScd}$ $$\begin{CD} \operatorname{Hom}_Q(Fp,q)@>>> \operatorname{Hom}_P(p,Gq)\\ @VVV @VVV \\ \operatorname{Hom}_Q(Fp',q)@>>> \operatorname{Hom}_P(p',Gq) \end{CD}$$ Since $\operatorname{Hom}_P(p',Gq)$ is the singleton there is a unique map into it, meaning that the two paths, from the upper-left vertex of the square to the lower-right one, coincide.
The naturality in one variable already implies that $F\dashv G$, but it's clear that you can get the naturalty in the second variable reasoning analogously.