In category theory, a functor is a mapping between categories. Practically speaking, this allows a function of type `A => B`

to be lifted to a function of type `F[A] => F[B]`

, for some type constructor `F`

.

```
.--------------. .-------------------.
| Category _ | | Category F[_] |
|--------------| |-------------------|
| A | | F[A] |
| B | | F[B] |
| A => B ~~~~~~~ map ~~~~> F[A] => F[B] |
'--------------' '-------------------'
```

This can be read as:

- The category
`_`

contains objects of type`A`

and`B`

, and a morphism from`A`

to`B`

. - The category
`F[_]`

contains objects of type`F[A]`

and`F[B]`

, and a morphism from`F[A]`

to`F[B]`

. - A functor for
`F[_]`

maps morphisms from`_`

, for example converting`A => B`

into`F[A] => F[B]`

.

This is useful when you have a value of type `F[A]`

and you want to apply a function of type `A => B`

to it, resulting in a value of type `F[B]`

.

```
trait Functor[F[_]] {
def map[A, B](fa: F[A])(f: A => B): F[B]
}
object Functor {
implicit class FunctorOps[A, F[_]: Functor](fa: F[A]) {
def map[B](f: A => B): F[B] =
implicitly[Functor[F]].map(fa)(f)
}
}
```