the direct product in Grp, or the product topology in Top, where products exist. Let and be objects of a category with finite products. The product of and is an object × together with two morphisms : : such that for any other object of and morphisms and there exists a unique morphism such that and . To understand this characterization as a universal property, take the category to be the product category and define the diagonal functor by and . Then is a universal morphism from to the object of : if is any morphism from to , then it must equal a morphism from to followed by . Limits and colimits Categorical products are a particular kind of limit in category theory. One can generalize the above example to arbitrary limits and colimits. Let and be categories with a small index category and let be the corresponding functor category. The diagonal functor is the functor that maps each object in to the constant functor (i.e. for each in and for each in ). Given a functor (thought of as an object in ), the limit of , if it exists, is nothing but a universal morphism from to . Dually, the colimit of is a universal morphism from to . Properties Existence and uniqueness Defining a quantity does not guarantee its existence. Given a functor and an object of , there may or may not exist a universal morphism from to . If, however, a universal morphism does exist, then it is essentially unique. Specifically, it is unique up to a unique isomorphism: if is another pair, then there exists a unique isomorphism such that . This is easily seen by substituting in the definition of a universal morphism. It is the pair which is essentially unique in this fashion. The object itself is only unique up to isomorphism. Indeed, if is a universal morphism and is any isomorphism then the pair , where is also a universal morphism. Equivalent formulations The definition of a universal morphism can be rephrased in a variety of ways. Let be a functor and let be an object of . Then the following statements are equivalent: is a universal morphism from to is an initial object of the comma category is a representation of The dual statements are also equivalent: is a universal morphism from to is a terminal object of the