Therefore, one strategy to prove that two objects are isomorphic is to show that they satisfy the same universal property. Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property. Universal properties occur everywhere in mathematics. By understanding their abstract properties, one obtains information about all these constructions and can avoid repeating the same analysis for each individual instance. Formal definition To understand the definition of a universal construction, it is important to look at examples. Universal constructions were not defined out of thin air, but were rather defined after mathematicians began noticing a pattern in many mathematical constructions (see Examples below). Hence, the definition may not make sense to one at first, but will become clear when one reconciles it with concrete examples. Let be a functor between categories and . In what follows, let be an object of , while and are objects of , and is a morphism in . Thus, the functor maps , and in to , and in . A universal morphism from to is a unique pair in which has the following property, commonly referred to as a universal property: For any morphism of the form in , there exists a unique morphism in such that the following diagram commutes: We can dualize this categorical concept. A universal morphism from to is a unique pair that satisfies the following universal property: For any morphism of the form in , there exists a unique morphism in such that the following diagram commutes: Note that in each definition, the arrows are reversed. Both definitions are necessary to describe universal constructions which appear in mathematics; but they also arise due to the inherent duality present in category theory. In either case, we say that the pair which behaves as above satisfies a universal property. Connection with comma categories Universal morphisms can be described more concisely as initial and terminal objects in a comma category (i.e. one where morphisms are seen as objects in their own right). Let be a functor and an object of . Then recall that the comma category is the category where Objects are pairs of the form , where is an object in A