In category theory, a branch of mathematics, a limit or a colimit of presheaves on a category C is a limit or colimit in the functor category .
The category admits small limits and small colimits. Explicitly, if is a functor from a small category I and U is an object in C, then is computed pointwise:
The same is true for small limits. Concretely this means that, for example, a fiber product exists and is computed pointwise.
When C is small, by the Yoneda lemma, one can view C as a full subcategory of . If is a functor, if is a functor from a small category I and if the colimit in is representable; i.e., isomorphic to an object in C, then, in D,
(in particular the colimit on the right exists in D). In other words, if the Yoneda embedding of C preserves the colimit of a functor into C, then every functor out of C preserves that colimit.
The density theorem states that every presheaf is a colimit of representable presheaves.