Because Set is invariant in the specified Scala library, even returning Set[U] would place the class's covariant type parameter in an invariant position. Introduce a fresh widened type for each lifting.
class CoPartialFnExt[T,+U](data: List[(T,U)])
    extends CoPartialFn[T,U](data) {

  def forwardLift[V >: U](s: Set[T]): Set[V] = {
    val ys: List[V] = this.data.collect {
      case (x,y) if s.contains(x) => y
    }
    ys.toSet
  }

  def backwardLift[V >: U](s: Set[V]): Set[T] =
    this.data.collect {
      case (x,y) if s.contains(y) => x
    }.toSet
}
The lower type bound gives , so stored output values can be widened into List[V] or passed to s.contains. Both exposed Set types use invariant method type parameters, not the class's U. Calling forwardLift[U] or backwardLift[U] when the concrete output type is known recovers the exact output universe; a broader admissible universe is also safe.
For a partial function, the forward formula is understood only for inputs in its domain of a function. The implementations therefore compute
(1)
The conversion to a set removes repeated images. Inputs not in the domain of a function contribute nothing to either lifting.

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