Free Monad with nested Coproduct

NestedCoproduct.scala

package sandbox

import cats.data.Coproduct
import cats.free.{Free, Inject}

sealed trait Alg1[A]
sealed trait Alg2[A]
sealed trait Alg3[A]

class Alg1Ops[F[_]](implicit I: Inject[Alg1, F]){
  def alg1: Free[F, Unit] = ???
}

class Alg2Ops[F[_]](implicit I: Inject[Alg2, F]){
  def alg2: Free[F, Unit] = ???
}

class Alg3Ops[F[_]](implicit I: Inject[Alg3, F]){
  def alg3: Free[F, Unit] = ???
}

object NestedCoproduct extends App {
  type API[A] = Coproduct[Alg1, Coproduct[Alg2, Alg3, ?], A]

  implicit def alg1Ops[F[_]](implicit I: Inject[Alg1, F]): Alg1Ops[F] = new Alg1Ops[F]()
  implicit def alg2Ops[F[_]](implicit I: Inject[Alg2, F]): Alg2Ops[F] = new Alg2Ops[F]()
  implicit def alg3Ops[F[_]](implicit I: Inject[Alg3, F]): Alg3Ops[F] = new Alg3Ops[F]()

//  compile if uncommented
//  Error:(40, 3) could not find implicit value for parameter A1: sandbox.Alg1Ops[sandbox.NestedCoproduct.API]
//  program
//  ^
//  implicit val inject3a: Inject[Alg1, API] = implicitly
  implicit val inject3b: Inject[Alg2, API] = implicitly
  implicit val inject3c: Inject[Alg3, API] = implicitly

  def program(implicit A1: Alg1Ops[API], A2: Alg2Ops[API], A3: Alg3Ops[API]): Free[API, Unit] =
    for {
      _ <- A1.alg1
      _ <- A2.alg2
      _ <- A3.alg3
    } yield ()

  program

}