IsoK.scala

trait IsoK[F[_], G[_]] { self =>
  def from[A](fa: F[A]): G[A]
  def to[A](ga: G[A]): F[A]

  def flip: IsoK[G, F] = new IsoK[G, F] {
    def from[A](ga: G[A]): F[A] = self.to(ga)
    def to[A](fa: F[A]): G[A] = self.from(fa)
  }

  def compose[H[_]](that: IsoK[G, H]): IsoK[F, H] = new IsoK[F, H]{
    def from[A](fa: F[A]): H[A] = that.from(self.from(fa))
    def to[A](ha: H[A]): F[A] = self.to(that.to(ha))
  }

  //convenient shortcut method
  implicit def monad(implicit F: Monad[F]): Monad[G] = IsoK.monad[F, G](F, this)
  implicit def applicative(implicit F: Applicative[F]): Applicative[G] = IsoK.applicative[F, G](F, this)
}

object IsoK extends IsoInstances with IsoProvidedInstances {
  def apply[F[_], G[_]](implicit inst: IsoK[F, G]): IsoK[F, G] = inst
}

trait IsoInstances {
  implicit def optionT[F[_]]: IsoK[OptionT[F, *], Lambda[A => F[Option[A]]]] = new IsoK[OptionT[F, *], Lambda[A => F[Option[A]]]] {
    def from[A](fa: OptionT[F, A]): F[Option[A]] = fa.value
    def to[A](ga: F[Option[A]]): OptionT[F, A] = OptionT(ga)
  }
}

sealed private[cats] trait IsoProvidedInstances {

  def applicative[F[_], G[_]](implicit F: Applicative[F], I: IsoK[F, G]): Applicative[G] =
    new Applicative[G] {
      import I._
      def pure[A](r: A): G[A] = from(F.pure(r))

      override def map[A, B](ga: G[A])(f: A => B): G[B] =
        from(F.map(to(ga))(f))

      override def map2[A, B, C](ga: G[A], gb: G[B])(fn: (A, B) => C): G[C] =
        from(F.map2(to(ga), to(gb))(fn))

      override def product[A, B](ga: G[A], gb: G[B]): G[(A, B)] =
        from(F.product(to(ga), to(gb)))

      def ap[A, B](f: G[A => B])(ga: G[A]): G[B] =
        from(F.ap(to(f))(to(ga)))
    }
  
  def monad[F[_], G[_]](implicit F: Monad[F], I: IsoK[F, G]): Monad[G] =
    new Monad[G] {

      import I._
      def pure[A](r: A): G[A] = from(F.pure(r))

      def flatMap[A, B](ga: G[A])(f: A => G[B]): G[B] =
        from(F.flatMap(to(ga))(f.map(to)))

      override def map[A, B](ga: G[A])(f: A => B): G[B] =
        from(F.map(to(ga))(f))

      override def map2[A, B, C](ga: G[A], gb: G[B])(fn: (A, B) => C): G[C] =
        from(F.map2(to(ga), to(gb))(fn))

      override def product[A, B](ga: G[A], gb: G[B]): G[(A, B)] =
        from(F.product(to(ga), to(gb)))

      override def ap[A, B](f: G[A => B])(ga: G[A]): G[B] =
        from(F.ap(to(f))(to(ga)))

      def tailRecM[A, B](a: A)(fn: A => G[Either[A, B]]): G[B] =
        from(F.tailRecM(a)(fn.map(to)))
    }

  def distributive[F[_], G[_]](implicit F: Distributive[F], I: IsoK[F, G]): Distributive[G] =
    new Distributive[G] {
      import I._
      def map[A, B](ga: G[A])(f: A => B): G[B] = from(F.map(to(ga))(f))
      def distribute[H[_]: Functor, A, B](ha: H[A])(f: A => G[B]): G[H[B]] =
        from(F.distribute(ha)(f.map(to)))
    }
}

Basically this allows you to easily take advantage of the Monad[Function1] if you alg trait is basically a A=>B or Transformer such as OptionT or Nested, if your alg trait is A => T[B]