∀ {R : Type u_4} {A : Type u_2} {B : Type u_1} [inst : CommSemiring R] [inst_1 : NonUnitalSemiring A]
  [inst_2 : StarRing A] [inst_3 : Module R A] [inst_4 : NonUnitalSemiring B] [inst_5 : StarRing B] [inst_6 : Module R B]
  {ι : Type u_3} [inst_7 : StarRing R] [inst_8 : IsScalarTower R A A] [inst_9 : SMulCommClass R A A]
  [inst_10 : StarModule R A] [inst_11 : Nonempty ι] (K : ι → NonUnitalStarSubalgebra R A)
  (dir : Directed (fun x1 x2 => x1 ≤ x2) K) (f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B)
  (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (NonUnitalStarSubalgebra.inclusion h)),
  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x) ⋯ ↑(iSup K) ⋯ 0 = 0