Syntax for function1 kleisli-composition

core/src/main/scala/cats/syntax/function1.scala

@@ -6,6 +6,9 @@ trait Function1Syntax {
   implicit def catsSyntaxFunction1[F[_]: Functor, A, B](fab: F[Function1[A, B]]): Function1Ops[F, A, B] =
     new Function1Ops[F, A, B](fab)
 
+  implicit def catsSyntaxFunction1FlatMap[F[_]: FlatMap, A, B](fab: Function1[A, F[B]]): Function1FlatMapOps[F, A, B] =
+    new Function1FlatMapOps[F, A, B](fab)
+
   final class Function1Ops[F[_]: Functor, A, B](fab: F[Function1[A, B]]) {
 
     /**
@@ -30,4 +33,57 @@ trait Function1Syntax {
      */
     def mapApply(a: A): F[B] = Functor[F].map(fab)(_(a))
   }
+
+  final class Function1FlatMapOps[F[_]: FlatMap, A, B](f: A => F[B]) {
+
+    /**
+     * Alias for `a => f(a).flatMap(g)` or `(Kleisli(f) andThen Kleisli(g)).run`
+     *
+     * Example:
+     * {{{
+     * scala> import scala.util._
+     * scala> import cats.implicits._
+     *
+     * scala> val f: List[String] => Option[String] = _.headOption
+     * scala> val g: String => Option[Int] = str => Try(str.toInt).toOption
+     * scala> (f >=> g)(List("42"))
+     * res0: Option[Int] = Some(42)
+     * scala> (f andThenF g)(List("abc"))
+     * res1: Option[Int] = None
+     * scala> (f andThenF g)(List())
+     * res2: Option[Int] = None
+     * }}}
+     */
+    def andThenF[C](g: B => F[C]): A => F[C] = a => FlatMap[F].flatMap(f(a))(g)
+
+    /**
+     * Alias for `f andThenF g`.
+     */
+    @inline def >=>[C](g: B => F[C]): A => F[C] = f.andThenF(g)
+
+    /**
+     * Alias for `c => g(c).flatMap(f)` or `(Kleisli(f) compose Kleisli(g)).run`
+     *
+     * Example:
+     * {{{
+     * scala> import scala.util._
+     * scala> import cats.implicits._
+     *
+     * scala> val f: String => Option[Int] = str => Try(str.toInt).toOption
+     * scala> val g: List[String] => Option[String] = _.headOption
+     * scala> (f composeF g)(List("42"))
+     * res0: Option[Int] = Some(42)
+     * scala> (f composeF g)(List("abc"))
+     * res1: Option[Int] = None
+     * scala> (f composeF g)(List())
+     * res2: Option[Int] = None
+     * }}}
+     */
+    def composeF[C](g: C => F[A]): C => F[B] = c => FlatMap[F].flatMap(g(c))(f)
+
+    /**
+     * Alias for `f composeF g`.
+     */
+    @inline def <=<[C](g: C => F[A]): C => F[B] = f.composeF(g)
+  }
 }

tests/src/test/scala/cats/tests/SyntaxSuite.scala

@@ -364,10 +364,20 @@ object SyntaxSuite {
     val fa = a.pure[F]
   }
 
-  def testFlatMap[F[_]: FlatMap, A, B]: Unit = {
+  def testFlatMap[F[_]: FlatMap, A, B, C, D]: Unit = {
     val a = mock[A]
     val returnValue = mock[F[Either[A, B]]]
     val done = a.tailRecM[F, B](a => returnValue)
+
+    val x = mock[Function[A, F[B]]]
+    val y = mock[Function[B, F[C]]]
+    val z = mock[Function[C, F[D]]]
+
+    val b = x.andThenF(y).andThenF(z)
+    val b2 = x >=> y >=> z
+
+    val c = z.composeF(y).composeF(x)
+    val c2 = z <=< y <=< x
   }
 
   def testApplicativeError[F[_, _], E, A, B](implicit F: ApplicativeError[F[E, *], E]): Unit = {