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Final functor

In category theory, the notion of final functor (resp. initial functor) is a generalization of the notion of final object (resp. initial object) in a category.

A functor F : C → D {\displaystyle F:C\to D} is called final if, for any set-valued functor G : D → Set {\displaystyle G:D\to {\textbf {Set}}} , the colimit of G is the same as the colimit of G ∘ F {\displaystyle G\circ F} . Note that an object d ∈ Ob(D) is a final object in the usual sense if and only if the functor { ∗ } → d D {\displaystyle \{*\}\xrightarrow {d} D} is a final functor as defined here.

The notion of initial functor is defined as above, replacing final by initial and colimit by limit.

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