-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ Functor.obj (F.sheafPushforwardContinuous RingCat J K) R
inst_3 :
  J.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_4 :
  K.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
X Y : Cᵒᵖ
f : X ⟶ Y
⊢ (unit S).val.map f ≫
      (ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).map
        (ModuleCat.homMk
          (CategoryTheory.Functor.map
            (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
              AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
              AddCommGrpCat.carrier))
            (φ.hom.app Y))
          ⋯) =
    ModuleCat.homMk
        (CategoryTheory.Functor.map
          (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
            AddCommGrpCat.carrier))
          (φ.hom.app X))
        ⋯ ≫
      ((pushforward φ).obj (unit R)).val.map f
have hf : Eq (α :=
  (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).obj
        (((pushforward φ).obj (unit R)).val.obj Y)) :
    Type u))
  ((ModuleCat.Hom.hom
          ((unit S).val.map f ≫
            (ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).map
              (ModuleCat.homMk
                (CategoryTheory.Functor.map
                  (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                    RingCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦
                    (↑x1 : Type u) →+ (↑x2 : Type u)) (CD := AddCommGrpCat.carrier))
                  (φ.hom.app Y))
                sorry)) :
        (a : (↑(S.obj.obj X) : Type u)) →
          (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj.map f))).obj
                (((pushforward φ).obj (unit R)).val.obj Y)) :
            Type u))
      1 :
    (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).obj
          (((pushforward φ).obj (unit R)).val.obj Y)) :
      Type u))
  (((ModuleCat.Hom.hom
            (ModuleCat.homMk
                (CategoryTheory.Functor.map
                  (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                    RingCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦
                    (↑x1 : Type u) →+ (↑x2 : Type u)) (CD := AddCommGrpCat.carrier))
                  (φ.hom.app X))
                sorry ≫
              ((pushforward φ).obj (unit R)).val.map f) :
          LinearMap (R := (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u))
            (RingHom.id (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u)) (↑((unit S).val.obj X) : Type u)
            (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).obj
                  (((pushforward φ).obj (unit R)).val.obj Y)) :
              Type u)) :
        (a : (↑(S.obj.obj X) : Type u)) →
          (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj.map f))).obj
                (((pushforward φ).obj (unit R)).val.obj Y)) :
            Type u))
      1 :
    (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).obj
          (((pushforward φ).obj (unit R)).val.obj Y)) :
      Type u))
Tactic: ext