category of sets is complete and cocomplete, and the identity functor is continuous and cocontinuous. Thus, the category of graphs is complete and cocomplete. The notion of a universal morphism to a particular colimit, or from a limit, can be expressed in terms of a comma category. Essentially, we create a category whose objects are cones, and where the limiting cone is a terminal object; then, each universal morphism for the limit is just the morphism to the terminal object. This works in the dual case, with a category of cocones having an initial object. For example, let be a category with the functor taking each object to and each arrow to . A universal morphism from to consists, by definition, of an object and morphism with the universal property that for any morphism there is a unique morphism with . In other words, it is an object in the comma category having a morphism to any other object in that category; it is initial. This serves to define the coproduct in , when it exists. Adjunctions Lawvere showed that the functors and are adjoint if and only if the comma categories and , with and the identity functors on and respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of . This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories. Natural transformations If the domains of are equal, then the diagram which defines morphisms in with is identical to the diagram which defines a natural transformation . The difference between the two notions is that a natural transformation is a particular collection of morphisms of type of the form , while objects of the comma category contains all morphisms of type of such form. A functor to the comma category selects that particular collection of morphisms. This is described succinctly by an observation by S.A. Huq that a natural transformation , with , corresponds to a functor which maps each object to and maps each morphism to . This is a bijective correspondence between natural transformations and functors which are sections of both forgetful functors from . References Lawvere, W (1963). "Functorial semantics of algebraic theories" and "Some algebraic problems in the context of functorial semantics of algebraic theories". http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf External links J. Adamek, H. Herrlich, G. Stecker, Abstract and