Preorder Sets Category

The Ord category has preordered sets as objects and increasing functions as morphisms. This is a category because the composition of two increasing functions is also increasing.

The monomorphisms in Ord are the increasing injective functions. The empty set (considered as an ordered set) is the initial object of Ord; any singleton is a terminal object.

The product in Ord is given by the product order in the cartesian product. The coproduct is given by the disjoint union of preordered sets.

We have a "forget" functor: Ord --> Set which assigns each preordered set the underlying set, and each increasing function the underlying function. This functor is faithful, and therefore Ord is a concrete category.