The identity monad can be seen as the ambient monad that encodes the effect of having no effect. It is ambient in the sense that plain pure values are values of Id.

It is encoded as:

type Id[A] = A

That is to say that the type Id[A] is just a synonym for A. We can freely treat values of type A as values of type Id[A], and vice-versa.

import cats._

val x: Id[Int] = 1
// x: Id[Int] = 1
val y: Int = x
// y: Int = 1

Using this type declaration, we can treat our Id type constructor as a Monad and as a Comonad. The pure method, which has type A => Id[A] just becomes the identity function. The map method from Functor just becomes function application:

import cats.Functor

val one: Int = 1
// one: Int = 1
Functor[Id].map(one)(_ + 1)
// res0: Id[Int] = 2

Compare the signatures of map and flatMap and coflatMap:

  def map[A, B](fa: Id[A])(f: A => B): Id[B]
  def coflatMap[A, B](a: Id[A])(f: Id[A] => B): Id[B]

You’ll notice that in the flatMap signature, since Id[B] is the same as B for all B, we can rewrite the type of the f parameter to be A => B instead of A => Id[B], and this makes the signatures of the two functions the same, and, in fact, they can have the same implementation, meaning that for Id, flatMap is also just function application:

import cats.Monad

val one: Int = 1
// one: Int = 1
Monad[Id].map(one)(_ + 1)
// res1: Id[Int] = 2
Monad[Id].flatMap(one)(_ + 1)
// res2: Id[Int] = 2

And that similarly, coflatMap is just function application:

import cats.Comonad

Comonad[Id].coflatMap(one)(_ + 1)
// res3: Id[Int] = 2