→ C be a diagram. A functor G : C → D is said to create limits for F if whenever (L, φ) is a limit of GF there exists a unique cone (L′, φ′) to F such that G(L′, φ′) = (L, φ), and furthermore, this cone is a limit of F. reflect limits for F if each cone to F whose image under G is a limit of GF is already a limit of F. Dually, one can define creation and reflection of colimits. The following statements are easily seen to be equivalent: The functor G creates limits. The functor G lifts limits uniquely and reflects limits. There are examples of functors which lift limits uniquely but neither create nor reflect them. Examples Every representable functor C → Set preserves limits (but not necessarily colimits). In particular, for any object A of C, this is true of the covariant Hom functor Hom(A,–) : C → Set. The forgetful functor U : Grp → Set creates (and preserves) all small limits and filtered colimits; however, U does not preserve coproducts. This situation is typical of algebraic forgetful functors. The free functor F : Set → Grp (which assigns to every set S the free group over S) is left adjoint to forgetful functor U and is, therefore, cocontinuous. This explains why the free product of two free groups G and H is the free group generated by the disjoint union of the generators of G and H. The inclusion functor Ab → Grp creates limits but does not preserve coproducts (the coproduct of two abelian groups being the direct sum). The forgetful functor Top → Set lifts limits and colimits uniquely but creates neither. Let Metc be the category of metric spaces with continuous functions for morphisms. The forgetful functor Metc → Set lifts finite limits but does not lift them uniquely. A note on terminology Older terminology referred to limits as "inverse limits" or "projective limits", and to colimits as "direct limits" or "inductive limits". This has been the source of a lot of confusion. There are several ways to remember the modern terminology. First of all, cokernels, coproducts, coequalizers, and codomains are types of colimits, whereas kernels, products equalizers, and domains are types of limits. Second, the prefix "co" implies "first variable of the ". Terms like "cohomology" and "cofibration" all have a slightly stronger association with