Category of preordered sets

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

The monomorphisms in Ord are the injective monotonic functions.

The empty set (considered as a preordered set) is the initial object of Ord; any singleton preordered set is a terminal object. There are thus no zero objects in Ord.

The product in Ord is given by the product order on the cartesian product.

We have a "forgetful" functor Ord → Set which assigns to each preordered set the underlying set, and to each monotonic function the underlying function. This functor is faithful, and therefore Ord is a concrete category.