-- Module: Mathlib.AlgebraicGeometry.Modules.Tilde
Goal:
R : CommRingCat
M_1 : ModuleCat (↑R : Type u)
M N : (Spec R).Modules
f : Functor.obj modulesSpecToSheaf M ⟶ Functor.obj modulesSpecToSheaf N
U : (Opens ↥(Spec R))ᵒᵖ
t :
  (↑(Functor.obj
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
        U) :
    Type u)
m : (↑(M.val.1 U) : Type u)
apply :
  ∀ x ∈ unop U,
    ∃ (V : Opens ↥(Spec R)) (hV : V ≤ unop U),
      x ∈ V ∧
        Eq (α :=
          (↑((Functor.obj modulesSpecToSheaf N).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).obj
                (op V)) :
            Type u))
          ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑R : Type u)) ↦
                  LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑x1 : Type u) (↑x2 : Type u)) (CC :=
                  ModuleCat.carrier)
                  ((Functor.obj modulesSpecToSheaf N).obj (P :=
                        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).map
                    (homOfLE hV).op) :
                (a : (↑((modulesSpecToSheaf.obj N).obj.obj (op (unop U))) : Type u)) →
                  (↑((modulesSpecToSheaf.obj N).obj.obj (op V)) : Type u))
              ((ModuleCat.Hom.hom (f.hom.app U)).toFun (t • m)) :
            (↑((Functor.obj modulesSpecToSheaf N).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).obj
                  (op V)) :
              Type u))
          ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑R : Type u)) ↦
                  LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑x1 : Type u) (↑x2 : Type u)) (CC :=
                  ModuleCat.carrier)
                  ((Functor.obj modulesSpecToSheaf N).obj (P :=
                        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).map
                    (homOfLE hV).op) :
                (a : (↑((modulesSpecToSheaf.obj N).obj.obj (op (unop U))) : Type u)) →
                  (↑((modulesSpecToSheaf.obj N).obj.obj (op V)) : Type u))
              (HSMul.hSMul (α :=
                (↑(Functor.obj
                      (ObjectProperty.FullSubcategory.obj (P :=
                        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
                      U) :
                  Type u))
                ((RingHom.id
                        (↑(Functor.obj
                              (ObjectProperty.FullSubcategory.obj (P :=
                                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R)))
                                (Spec R).ringCatSheaf)
                              U) :
                          Type u) :
                      (a : (↑((Spec R).ringCatSheaf.obj.obj U) : Type u)) →
                        (↑((Spec R).ringCatSheaf.obj.obj U) : Type u))
                    t :
                  (↑(Functor.obj
                        (ObjectProperty.FullSubcategory.obj (P :=
                          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
                        U) :
                    Type u))
                ((ModuleCat.Hom.hom (f.hom.app U)).toFun m)) :
            (↑((Functor.obj modulesSpecToSheaf N).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).obj
                  (op V)) :
              Type u))
⊢ (ModuleCat.Hom.hom (f.hom.app U)).toFun (t • m) =
    HSMul.hSMul (α :=
      (↑(Functor.obj
            (ObjectProperty.FullSubcategory.obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
            U) :
        Type u))
      ((RingHom.id
              (↑(Functor.obj
                    (ObjectProperty.FullSubcategory.obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
                    U) :
                Type u) :
            (a : (↑((Spec R).ringCatSheaf.obj.obj U) : Type u)) → (↑((Spec R).ringCatSheaf.obj.obj U) : Type u))
          t :
        (↑(Functor.obj
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
              U) :
          Type u))
      ((ModuleCat.Hom.hom (f.hom.app U)).toFun m)

Tactic:
apply TopCat.Presheaf.IsSheaf.section_ext (modulesSpecToSheaf.obj N).2

Premises used: TopCat.Presheaf.IsSheaf.section_ext, CategoryTheory.ObjectProperty.FullSubcategory.property, AlgebraicGeometry.modulesSpecToSheaf, CategoryTheory.Functor.obj