Sorry, I don't understand your notation. To me a monad is defined by two functions, pure and bind (aka flatMap). I don't understand how — × M corresponds to this.

— × M is the functor. In Haskell it would generally be used as "Writer M", but (M,) can also be used.

Sorry, that doesn't help. I literally do not understand the notation. I do not know what — means in this context, I don't understand what role × plays here, etc.

"—" means "the parameter of the type constructor goes here". I believe in Scala you use underscore for this, at the value level at least. "×" is what mathematicians call the type constructor for the usual product type. In Haskell this is called (,). I don't know what it's called in Scala.

So, — × M means something like "type T a = T a M", in Haskell notation.

In your notation it would be something like

  final case class Mon[A](v: A, m: M) {
    def flatMap[B](f: A => Mon[B]): Mon[B] =
      Mon(f(this.v).v, f(this.v).m * this.m)  // * is M's binary operation
  }
  object Mon {
    def pure(v: A): Mon[A] =
      Mon(v, M.unit)  // M.unit is M's identity element
  }