---
tag: M5910N7YF0
kind: concept
name: Final functor
msc: [18A30]
sources:
  - url: https://en.wikipedia.org/wiki/Final_functor
same_as:
  wikidata: Q18207102
  wikipedia: Final functor
---

## Summary

A final functor $F \colon \mathcal{C} \to \mathcal{D}$ is a [[functor|28NQ0KRHD3]] such that restricting a diagram on $\mathcal{D}$ along $F$ does not change its [[colimit|BJS1CGTKQ5]].
It is the categorical form of a cofinal subset of a [[directed set|D9RY4667K6]], as in [[cofinality|BAWHE1W6MH]].
Final functors let one compute a colimit over a smaller or simpler index category.
Some authors call them cofinal functors; the dual notion is an initial functor.

## Explanations

### L1 human

To find the limit of a sequence, you may look only at its terms with even index, since they go just as far out.
A final functor is the same idea for colimits in a category.
For example, if the index category has a terminal object, the colimit of any diagram is simply its value at that object.
This is because the functor that picks out the terminal object is final.