/-- Define a non-unital star algebra homomorphism on a directed supremum of non-unital star
subalgebras by defining it on each non-unital star subalgebra, and proving that it agrees on the
intersection of non-unital star subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalStarSubalgebra R A)
    (dir : Directed (· ≤ ·) K) (f : ∀ i, K i →⋆ₙₐ[R] B)
    (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))
    (T : NonUnitalStarSubalgebra R A) (hT : T = iSup K) : ↥T →⋆ₙₐ[R] B := by
  subst hT
  exact
    { toFun :=
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)
          (fun i j x hxi hxj => by
            let ⟨k, hik, hjk⟩ := dir i j
            simp only
            rw [hf i k hik, hf j k hjk]
            rfl)
          (↑(iSup K)) (by rw [coe_iSup_of_directed dir])
      map_zero' := by
        dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast]
        exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl)  _ (by simp)
      map_mul' := by
        dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast, ZeroMemClass.coe_zero,
          AddSubmonoid.mk_add_mk, Set.inclusion_mk]
        apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))
        on_goal 3 => rw [coe_iSup_of_directed dir]
        all_goals simp
      map_add' := by
        dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast]
        apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))
        

Proof state:
case hopi
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x + y) = (f i) x + (f i) y
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)

Next tactic:
on_goal 3 => rw [coe_iSup_of_directed dir]