## Functor

A functor is a mapping between two categories. Let \(\C\) and \(\D\) be two categories. A functor \(\F\) from \(\C\) to \(\D\):

- Associate objects: \(A\in\ob{\C}\) to \(\F(A)\in\ob{\D}\)
- Associate morphisms: \(f:A\to B\) to \(\F(f) : \F(A) \to \F(B)\) such that
  - \( \F (\)\(\id_X\)\()= \)\(\id\)\(\vphantom{\id}_{\F(}\)\(\vphantom{\id}_X\)\(\vphantom{\id}_{)} \),
  - \( \F (\)\(g∘f\)\()= \)\( \F(\)\(g\)\() \)\(\circ\)\( \F(\)\(f\)\() \)