Kleisli composition is to flatMap as function composition is to map.

gistfile1.scala

scala> import scalaz._
import scalaz._

scala> import Scalaz._
import Scalaz._

scala> val a = (value: Int) => value * 2
a: Int => Int = <function1>

scala> val b = (value: Int) => value + 1
b: Int => Int = <function1>

scala> 1.some.map(a).map(b)
res0: Option[Int] = Some(3)

// Compose the functions together.
scala> val ab = a.andThen(b)
ab: Int => Int = <function1>

// Is the same as "1.some.map(a).map(b)".
scala> 1.some.map(ab)
res1: Option[Int] = Some(3)

scala> val c = Kleisli((value: Int) => (value * 2).some)
c: scalaz.Kleisli[Option,Int,Int] = scalaz.KleisliFunctions$$anon$18@6b997837

scala> val d = Kleisli((value: Int) => (value + 1).some)
d: scalaz.Kleisli[Option,Int,Int] = scalaz.KleisliFunctions$$anon$18@6f3ee250

scala> 1.some.flatMap(c).flatMap(d)
res2: Option[Int] = Some(3)

// Compose the functions together.
scala> val cd = c.andThen(d)
cd: scalaz.Kleisli[Option,Int,Int] = scalaz.KleisliFunctions$$anon$18@25e9a396

// Is the same as "1.some.flatMap(c).flatMap(d)".
scala> 1.some.flatMap(cd)
res3: Option[Int] = Some(3)