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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 C ^ = F c t {\displaystyle {\widehat {C}}=\mathbf {Fct} }.
The category C ^ {\displaystyle {\widehat {C}}} admits small limits and small colimits. Explicitly, if f : I → C ^ {\displaystyle f:I\to {\widehat {C}}} is a functor from a small category I and U is an object in C, then lim → i ∈ I f {\displaystyle \varinjlim _{i\in I}f} 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 the full subcategory of C ^ {\displaystyle {\widehat {C}}}. If η : C → D {\displaystyle \eta :C\to D} is a functor, if f : I → C {\displaystyle f:I\to C} is a functor from a small category I and if the colimit lim → f {\displaystyle \varinjlim f} in C ^ {\displaystyle {\widehat {C}}} is representable; i.e., isomorphic to an object in C, then, in D,