∀ (X : CategoryTheory.Limits.WalkingParallelPair),
  ((CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.WalkingParallelFamily.Hom.line { down := true })
              (CategoryTheory.Limits.WalkingParallelFamily.Hom.line { down := false })).comp
          (CategoryTheory.Limits.parallelFamily fun p =>
            bif p.down then CategoryTheory.Limits.WalkingParallelPairHom.left
            else CategoryTheory.Limits.WalkingParallelPairHom.right)).obj
      X =
    (CategoryTheory.Functor.id CategoryTheory.Limits.WalkingParallelPair).obj X