Monad Transformer typeclass

I saw this: https://github.com/typelevel/cats/pull/4724 and wondered if we have a monad transformer typeclass similar to: https://hackage.haskell.org/package/transformers-0.6.1.2/docs/Control-Monad-Trans-Class.html

I don't see one. I can imagine something like:

trait MonadTrans[T[_[_], _]] {
  def liftK[M[_]: Monad]: FunctionK[M, [A] =>> T[M, A]]
  def lift[M[_]: Monad, A](ma: M[A]): T[M, A]

  implicit def transMonad[M[_]: Monad]: Monad[[A] =>> T[M, A]]
}

And possibly even:

trait MonadTrans[T[_[_], _]] {
  def liftK[M[_]: Monad]: FunctionK[M, [A] =>> T[M, A]]
  def lift[M[_]: Monad, A](ma: M[A]): T[M, A]
  
  implicit def transMonad[M[_]: Monad]: Monad[[A] =>> T[M, A]]
  
  type RunT[_]
  def run[M[_]: Monad, A](fa: T[M, A]): M[RunT[A]]
  def liftRun[M[_]: Monad, A](run: M[RunT[A]]): T[M, A]
}

since I think all the transformers seem to have some notion of a "run" version. The laws would be:

MonadTrans[T].lift(fa) == MonadTrans[T].liftK[M](fa)

MonadTrans[T].lift(Monad[M].pure(a)) == MonadTrans[T].transMonad[M].pure(a)

MonadTrans[T].lift(ma.flatMap(f)) == MonadTrans[T].lift(ma).flatMap(a => MonadTrans[T].lift(f(a)))

MonadTrans[T].liftRun(MonadTrans[T].run(fa)) == fa

The argument against this, I think, would be that it's not clear what useful functions you would write against MonadTrans except perhaps for having a clear way to lift: MonadTrans[OptionT].lift(fa)

Start by reviewing the proposed MonadTrans signatures and laws in this issue, then compare them with Cats pull request 4724 and Haskell's transformers MonadTrans documentation. No repository files or tests are named; done would require an agreed design and a clear implementation and test scope.