Proposition (products in the category of preordered sets):
Let be a family of preordered sets. Then a product of this family in the category is given by the set together with the product order, where the projections are given by the functions
Proof: The functions are order homomorphisms, because if and , then by definition. Thus, we have a cone. Moreover, if the order homomorphisms define another cone, then as for the set product, the function given by
is the unique function from to such that for all , and it is an order homomorphism, because if , then for all we have .
Proposition (coproducts in the category of preordered sets):
Let be a family of preordered sets. Then a coproduct of this family in the category is given by the parallel composition of the , where the inclusions are given by the functions
Proof: The are order homomorphisms by definition of the parallel order, so that we do have a cocone. Suppose now that the maps define another cocone in the category . Then the unique function such that for all is given by
as in set theory, and it is an order homomorphism because if , then by def. of the parallel order and consequently .
Proposition (products and coproducts in the category of posets):
In the category , products and coproducts are given by the respective products and coproducts in the category .