nLab
PreOrd

PreOrdPreOrd denotes the category of preorders and order-preserving maps.

The category PreOrdPreOrd is topological over SetSet and therefore enjoys strong completeness and cocompleteness properties; see total category. It is also a locally presentable category.

PreOrdPreOrd is a cartesian closed category. In fact it is an exponential ideal in the cartesian closed category Cat.