Let $\mathbf{Po}$ be the category of posets with monotone maps, $\mathbf{PrSp}$ the category of Priestley spaces. Then let $U:\mathbf{PrSp}\to \mathbf{Po}$ be the forgetful functor. The profinite completion functor $P:\mathbf{Po}\to\mathbf{PrSp}$ would be the left adjoint of $U$. Let $\mathbf{DLat}$ be the category of bounded distributive lattices. Now by Birkhoff duality $\mathbf{DLat}_\text{fin}$ is dual to $\mathbf{Po}_\text{fin}$ and Priestley duality is that $\mathbf{DLat}=\text{Ind-}\mathbf{DLat}_\text{fin}$ is dual to $\mathbf{PrSp}=\text{Pro-}\mathbf{Po}_\text{fin}$. Let $D:\mathbf{Po}\to \mathbf{DLat}$ be the contravariant functor sending a po $P$ to the lattice $\mathcal{D}(P)$ of its downsets, and a monotone map $f:P\to Q$ to the preimage map $f^{-1}:\mathcal{D}(Q)\to\mathcal{D}(P)$. Let $H:\mathbf{DLat}\to \mathbf{PrSp}$ be the contravariant functor in Priestley duality. Now I claim that $P:\mathbf{Po}\to \mathbf{PrSp}$ can be defined as $P=H\circ D$.

## Sketch of the proof

Let $Q$ be a partial order and $\langle p_{i,j}:Q_i\to Q_j \rangle$ the cofiltered diagram of its finite po quotients. Let $P(Q)=\lim_i Q_i$ be the limit in the category of Priestley spaces, each $Q_i$ considered discrete of course, and $i_Q:Q\to P(Q)$ the natural monotone mapping. Using Birkhoff duality, we may consider the diagram $\langle h_j,i:D_j\to D_i\rangle$ in $\mathbf{DLat}_\text{fin}$ dual to $\langle p_i:Q\to Q_i \rangle$ and its colimit $D=\text{colim}_j D_j$ in $\mathbf{DLat}$. This distributive lattice is just the lattice $\mathcal{D}(Q)$ of downsets of $Q$. Indeed $D$ is the union of all downsets of $Q$ of the form $p^{-1}(E)$ for a monotone map $p:Q\to F$ with $F$ finite and $E$ downset of $F$. But for every downset $L$ of $Q$ the characteristic function $\chi_L:Q\to 2$ into $2=\{1<0\}$ is monotone and $\chi_L^{-1}(\{1\})=L$. It follows by Priestley duality that the Priestley space $P(Q)=\lim_i Q_i$ is dual to the lattice $\mathcal{D}(Q)$. From this point of view we get that $i_Q:Q\to P(Q)$, $q\mapsto \{D\in \mathcal{D}(Q)\mid q\in D\}$ is the unit of the adjunction.

Now let $f:Q\to X$ be a monotone map into a Priestley space $X$. Then $X$ is dual to the lattice $\mathcal{CD}(X)$ of clopen downsets of $X$ under Priestley duality. Since $f$ is monotone the preimage map goes $f^{-1}:\mathcal{D}(X)\to\mathcal{D}(Q)$ and restricts to a homomorphism $f^{-1}:\mathcal{CD}(X)\to\mathcal{D}(Q)$ whose Priesltey dual is a monotone continuous map $\bar{f}:P(Q)\to X$. Clearly we have $\bar{h}\circ i_Q=h$. Moreover $\bar{h}$ is the unique Priestley map from $P(Q)\to X$ with this property since $Q$ is dense in $P(Q)$.

Hence the functor $P:\mathbf{Po}\to\mathbf{PrSp}$, $Q\mapsto P(Q)$ and for a monotone $f:Q\to R$ in $\mathbf{Po}$ then $P(f):P(Q)\to P(R)$ is the Priestley dual map of $f^{-1}:\mathcal{D}(R)\to\mathcal{D}(Q)$, is left adjoint to the forgetful functor $U:\mathbf{PrSp}\to \mathbf{Po}$.

1more comment