In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious embedding from finite partial orders into Priestley spaces is the pro-completion of finite partial orders.

Main question: Does it follow that there exists a profinite completion functor from partial orders to Priestley spaces?

If yes, is there an explicit construction of the profinite completion of a partial order?

For example, does anyone know what does the profinite completion of the partial order $(\omega,=)$ look like?

Moreover, profinite quasi-orders are the the Priestley quasi-orders, i.e. quasi-ordered Stone spaces such that if $x\not\leq y$ then there is a clopen downset $D$ such that $y\in D$ and $x\not\in D$. I have the same questions about the profinite completion of a quasi-order. In this context, does one gain anything by considering quasi-order instead of partial orders?

For the record I asked this question on MathSE, here.

My try: The category $\mathbf{Qo}$ of quasi-orders with monotone maps is complete. The inclusion of $\mathbf{Qo}_\text{fin}$, the finite quasi orders, into $\text{Pro-}\mathbf{Qo}_\text{fin}$ extends to the functor $U:\text{Pro-}\mathbf{Qo}_\text{fin}\to \mathbf{Qo}$, bringing a formal directed limit $\lim_i Q_i$ of finite qo's to the actual limit of this diagram in $\mathbf{Qo}$. Since $\text{Pro-}\mathbf{Qo}_\text{fin}$ is equivalent to $\mathbf{PrQo}$ the category of Priestley qos, if I am not mistaken, the above functor $U:\mathbf{PrQo}\to\mathbf{Qo}$ is simply the forgetful functor. The profinite completion functor $P:\mathbf{Qo}\to\mathbf{PrQO}$ would be the left adjoint of $U$. Can we use some adjoint functor theorem to conlcude?