lim is a left inverse of Δ. This fails if J is not connected. For example, if J is a discrete category, the components of the unit are the diagonal morphisms δ : N → NJ. Dually, if every diagram of shape J has a colimit in C (for J small) there exists a colimit functor which assigns each diagram its colimit. This functor is left adjoint to the diagonal functor Δ : C → CJ, and one has a natural isomorphism The unit of this adjunction is the universal cocone from F to colim F. If J is connected (and nonempty) then the counit is an isomorphism, so that colim is a left inverse of Δ. Note that both the limit and the colimit functors are covariant functors. As representations of functors One can use Hom functors to relate limits and colimits in a category C to limits in Set, the category of sets. This follows, in part, from the fact the covariant Hom functor Hom(N, –) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits. If a diagram F : J → C has a limit in C, denoted by lim F, there is a canonical isomorphism which is natural in the variable N. Here the functor Hom(N, F–) is the composition of the Hom functor Hom(N, –) with F. This isomorphism is the unique one which respects the limiting cones. One can use the above relationship to define the limit of F in C. The first step is to observe that the limit of the functor Hom(N, F–) can be identified with the set of all cones from N to F: The limiting cone is given by the family of maps πX : Cone(N, F) → Hom(N, FX) where X(ψ) = ψX. If one is given an object L of C together with a natural isomorphism Φ : Hom(L, –) → Cone(–, F), the object L will be a limit of F with the limiting cone given by ΦL(idL). In fancy language, this amounts to saying that a limit of F is a representation of the functor Cone(–, F) : C → Set. Dually, if a diagram F : J → C has a colimit in C, denoted colim F, there is a unique canonical isomorphism which is natural in the variable N and respects the