An inverse Fraïssé limit for finite posets and duality for posets and lattices

We consider the category of all finite partial orderings with quotient maps as arrows and construct the Fraïssé sequence in this category. Then we use well known relations between partial orders and lattices to construct a sequence of lattices associated with the Fraïssé sequence. Each of these two sequences has a limit object – an inverse limit, which is also an object of our interest.