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, what is known about this profinite completion of partial orders?
For example, does anyone know what does the profinite completion of the partial order $(\omega,=)$ look like?
Does one gain anything from considering quasi-orders (also called preorders) instead of partial orders?
I asked this question more precisely on MathOverflow: here.