Basic constructions in the categories of sets, sets with a binary relation on them, preorders, and posets

The purpose of this note is to work out the details of the concrete incarnation of a few categorical constructions (products, coproducts, pullbacks, pushouts, equalizers, coequalizers, and exponentials) in some useful and basic categories: the categories of sets, sets endowed with a binary relation, preorders, and posets.