Universal Property

A characterization of an object by a universal mapping condition: an object U equipped with a universal arrow (or cone/cocone) to or from a diagram such that every other arrow into/from the diagram factors uniquely through U up to unique isomorphism, thereby specifying U up to unique isomorphism rather than by construction.

A characterization of an object or morphism in a category by a universal mapping condition: it is defined up to unique isomorphism by the property that maps into or out of it correspond naturally to maps into or out of every other object with the same property.