Alternative syntax for polymorphic lambdas (for Shapeless)

Currently kind-projector allows

val g = λ[Id ~> Option].run(x => Some(x))

to be rewritten as

val g = new (Id ~> Option) {
  def run[A](x: Id[A]): Option[A] = Some(x)
}

I was wondering if there could be an alternative syntax that rewrites to

object g extends (Id ~> Option) {
  def apply[A](x: Id[A]): Option[A] = Some(x)
  
// if possible it should allow rewrite to other method names as well:
//  def run[A](x: Id[A]): Option[A] = Some(x)
}

in situations where the singleton type g.type is required.

i.e. in Shapeless

the[Case1.Aux[g.type, Id[Int], Option[Int]]]

should compile.

Start by examining the existing λ[Id => Option].run syntax and the requested object g extension syntax. Use the Shapeless Case1.Aux[g.type, Id[Int], Option[Int]] example as the acceptance case, and determine whether alternate method names such as run must also be supported. Done means the singleton g.type form compiles in the stated use case.