-- Lean 4 proof step from Mathlib.AlgebraicGeometry.AffineScheme (goal)
Goal:
X Y : Scheme
U_1 : X.Opens
hU_1 : IsAffineOpen U_1
f_1 : (↑Γ(X, U_1) : Type u)
f : X ⟶ Y
x : ↥X
U : Y.Opens
hU : IsAffineOpen U
V : X.Opens
hV : IsAffineOpen V
hVU : V ≤ f ⁻¹ᵁ U
hx : x ∈ V
this_1 : Algebra (↑(Y.presheaf.obj (op U)) : Type u)
  (↑(Y.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯) : ↥Y)) : Type u) :=
  Y.presheaf.algebra_section_stalk (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)
this_2 :
  IsLocalization.AtPrime (↑(Y.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯) : ↥Y)) : Type u)
    (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal
this_3 : Algebra (↑(X.presheaf.obj (op V)) : Type u)
  (↑(X.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx) : ↥X)) : Type u) :=
  X.presheaf.algebra_section_stalk (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)
this :
  IsLocalization.AtPrime (↑(X.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx) : ↥X)) : Type u)
    (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal
refine_1 :
  Y.presheaf.stalk (f x) ≅
    CommRingCat.of (Localization.AtPrime (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal)
refine_2 :
  X.presheaf.stalk x ≅
    CommRingCat.of (Localization.AtPrime (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal)
⊢ Arrow.mk (Scheme.Hom.stalkMap f x) ≅
    Arrow.mk (X :=
      CommRingCat.of (Localization.AtPrime (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal))
      (Y :=
      CommRingCat.of (Localization.AtPrime (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal))
      (CommRingCat.ofHom
        (Localization.localRingHom (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal
          (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal
          (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU)) ⋯))
have refine_3 : Eq (α :=
  Y.presheaf.stalk (f x) ⟶
    CommRingCat.of (Localization.AtPrime (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal))
  (CategoryStruct.comp (Y :=
    CommRingCat.of (Localization.AtPrime (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) sorry)).asIdeal))
    (Iso.hom
      (RingEquiv.toCommRingCatIso
        (AlgEquiv.toRingEquiv
          (IsLocalization.algEquiv (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) sorry)).asIdeal.primeCompl
            (↑(Y.presheaf.stalk (f x)) : Type u)
            (Localization.AtPrime (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) sorry)).asIdeal)))))
    (CommRingCat.ofHom
      (Localization.localRingHom (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) sorry)).asIdeal
        (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal
        (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU)) sorry)))
  (Scheme.Hom.stalkMap f x ≫
    Iso.hom
      (RingEquiv.toCommRingCatIso
        (AlgEquiv.toRingEquiv
          (IsLocalization.algEquiv (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal.primeCompl
            (↑(X.presheaf.stalk x) : Type u)
            (Localization.AtPrime (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal)))))
Tactic: refine Arrow.isoMk' _ _ ?_ ?_ ?_