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

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