for the existence of all (co)limits of shape J. Universal property Limits and colimits are important special cases of universal constructions. Let C be a category and let J be a small index category. The functor category CJ may be thought of as the category of all diagrams of shape J in C. The diagonal functor is the functor that maps each object N in C to the constant functor Δ(N) : J → C to N. That is, Δ(N)(X) = N for each object X in J and Δ(N)(f) = idN for each morphism f in J. Given a diagram F: J → C (thought of as an object in CJ), a natural transformation ψ : Δ(N) → F (which is just a morphism in the category CJ) is the same thing as a cone from N to F. To see this, first note that Δ(N)(X) = N for all X implies that the components of ψ are morphisms ψX : N → F(X), which all share the domain N. Moreover, the requirement that the cone's diagrams commute is true simply because this ψ is a natural transformation. (Dually, a natural transformation ψ : F → Δ(N) is the same thing as a co-cone from F to N.) Therefore, the definitions of limits and colimits can then be restated in the form: A limit of F is a universal morphism from Δ to F. A colimit of F is a universal morphism from F to Δ. Adjunctions Like all universal constructions, the formation of limits and colimits is functorial in nature. In other words, if every diagram of shape J has a limit in C (for J small) there exists a limit functor which assigns each diagram its limit and each natural transformation η : F → G the unique morphism lim η : lim F → lim G commuting with the corresponding universal cones. This functor is right adjoint to the diagonal functor Δ : C → CJ. This adjunction gives a bijection between the set of all morphisms from N to lim F and the set of all cones from N to F which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected (and nonempty) then the unit of the adjunction is an isomorphism so that