-- 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))
M : SheafOfModules R
f : unit R ⟶ M
X : Cᵒᵖ
this :
  Eq (α := (↑(((pushforward φ).obj (unit R)).val.obj X) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u)) ↦
            LinearMap (R := (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u))
              (RingHom.id (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u)) (↑x1 : Type u) (↑x2 : Type u))
            (CC := ModuleCat.carrier) (X := (PresheafOfModules.unit (S.obj (P := Presheaf.IsSheaf J))).obj X) (Y :=
            ((pushforward φ).obj (unit R)).val.obj X)
            (PresheafOfModules.Hom.app (Hom.val (unitToPushforwardObjUnit φ)) X :
              (PresheafOfModules.unit (S.obj (P := Presheaf.IsSheaf J))).obj X ⟶
                ((pushforward φ).obj (unit R)).val.obj X) :
          (a : (↑((PresheafOfModules.unit S.obj).obj X) : Type u)) →
            (↑(((pushforward φ).obj (unit R)).val.obj X) : Type u))
        1 :
      (↑(((pushforward φ).obj (unit R)).val.obj X) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            (φ.hom.app X) :
          (a : (↑(S.obj.obj X) : Type u)) → (↑(R.obj.obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
        1 :
      (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
⊢ Eq (α := (↑(((pushforward φ).obj M).val.obj X) : Type u))
    ((ConcreteCategory.hom (FC :=
            fun
              (x1 x2 :
                ModuleCat (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u)) ↦
            LinearMap (R := (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
              (RingHom.id (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
              (↑x1 : Type u) (↑x2 : Type u))
            (CC := ModuleCat.carrier) (f.val.app (Opposite.op (F.obj (Opposite.unop X)))) :
          (a : (↑((PresheafOfModules.unit R.obj).obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u)) →
            (↑(M.val.obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
        1 :
      (↑(M.val.obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u)) ↦
            LinearMap (R := (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u))
              (RingHom.id (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u)) (↑x1 : Type u) (↑x2 : Type u))
            (CC := ModuleCat.carrier) (X := (PresheafOfModules.unit (S.obj (P := Presheaf.IsSheaf J))).obj X)
            (PresheafOfModules.Hom.app
                (Hom.val (unitToPushforwardObjUnit φ) ≫ (PresheafOfModules.pushforward φ.hom).map f.val) X :
              (PresheafOfModules.unit (S.obj (P := Presheaf.IsSheaf J))).obj X ⟶ ((pushforward φ).obj M).val.obj X) :
          (a : (↑((PresheafOfModules.unit S.obj).obj X) : Type u)) → (↑(((pushforward φ).obj M).val.obj X) : Type u))
        1 :
      (↑(((pushforward φ).obj M).val.obj X) : Type u))
have h : Eq (α := (↑(((pushforward φ).obj M).val.obj X) : Type u))
  ((ConcreteCategory.hom (FC :=
          fun
            (x1 x2 :
              ModuleCat (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u)) ↦
          LinearMap (R := (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
            (RingHom.id (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
            (↑x1 : Type u) (↑x2 : Type u))
          (CC := ModuleCat.carrier) (f.val.app (Opposite.op (F.obj (Opposite.unop X)))) :
        (a : (↑((PresheafOfModules.unit R.obj).obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u)) →
          (↑(M.val.obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
      1 :
    (↑(M.val.obj (Opposite.op (F.obj (Opposite.unop X)))) : Type u))
  ((ModuleCat.Hom.hom (((PresheafOfModules.pushforward φ.hom).map f.val).app X) :
        (a : (↑(((pushforward φ).obj (unit R)).val.obj X) : Type u)) → (↑(((pushforward φ).obj M).val.obj X) : Type u))
      1 :
    (↑(((pushforward φ).obj M).val.obj X) : Type u))
Tactic: simp +instances [this, map_one]