-- Module: Mathlib.AlgebraicGeometry.Modules.Tilde
Goal:
R : CommRingCat
M_1 : ModuleCat (↑R : Type u)
M : (Spec (CommRingCat.of (↑R : Type u))).Modules
f g : (↑R : Type u)ᵒᵖ
i : f ⟶ g
N : ModuleCat (↑(CommRingCat.of (↑R : Type u)) : Type u) := (Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)
hf :
  ∀ (x : (↑((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) : Type u)),
    Eq (α :=
      (↑((Functor.obj modulesSpecToSheaf M).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
            (op (PrimeSpectrum.basicOpen (unop g)))) :
        Type u))
      ((ModuleCat.Hom.hom
              (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤))
                  (PrimeSpectrum.basicOpen (unop g)) ≫
                ModuleCat.ofHom
                  (IsLocalizedModule.lift (Submonoid.powers (unop g))
                    (ModuleCat.Hom.hom
                      (tilde.toOpen
                        ((Functor.obj modulesSpecToSheaf M).obj (P :=
                              CategoryTheory.Presheaf.IsSheaf
                                (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
                          (op ⊤))
                        (PrimeSpectrum.basicOpen (unop g))))
                    (ModuleCat.Hom.hom
                      ((Functor.obj modulesSpecToSheaf M).obj (P :=
                            CategoryTheory.Presheaf.IsSheaf
                              (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
                        (Quiver.Hom.op (homOfLE ⋯))))
                    ⋯)) :
            (a : (↑((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)) : Type u)) →
              (↑((modulesSpecToSheaf.obj M).obj.obj (op (PrimeSpectrum.basicOpen (unop g)))) : Type u))
          x :
        (↑((Functor.obj modulesSpecToSheaf M).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf
                    (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
              (op (PrimeSpectrum.basicOpen (unop g)))) :
          Type u))
      ((ModuleCat.Hom.hom
              (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤))
                  (PrimeSpectrum.basicOpen (unop f)) ≫
                ModuleCat.ofHom
                    (IsLocalizedModule.lift (Submonoid.powers (unop f))
                      (ModuleCat.Hom.hom
                        (tilde.toOpen
                          ((Functor.obj modulesSpecToSheaf M).obj (P :=
                                CategoryTheory.Presheaf.IsSheaf
                                  (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
                            (op ⊤))
                          (PrimeSpectrum.basicOpen (unop f))))
                      (ModuleCat.Hom.hom
                        ((Functor.obj modulesSpecToSheaf M).obj (P :=
                              CategoryTheory.Presheaf.IsSheaf
                                (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
                          (Quiver.Hom.op (homOfLE ⋯))))
                      ⋯) ≫
                  (Functor.obj modulesSpecToSheaf M).obj (P :=
                        CategoryTheory.Presheaf.IsSheaf
                          (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
                    i.unop.hom.op) :
            (a : (↑((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)) : Type u)) →
              (↑((modulesSpecToSheaf.obj M).obj.obj (op (PrimeSpectrum.basicOpen (unop g)))) : Type u))
          x :
        (↑((Functor.obj modulesSpecToSheaf M).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf
                    (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
              (op (PrimeSpectrum.basicOpen (unop g)))) :
          Type u))
⊢ ModuleCat.Hom.hom
      (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) (PrimeSpectrum.basicOpen (unop g)) ≫
        ModuleCat.ofHom
          (IsLocalizedModule.lift (Submonoid.powers (unop g))
            (ModuleCat.Hom.hom
              (tilde.toOpen
                ((Functor.obj modulesSpecToSheaf M).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
                  (op ⊤))
                (PrimeSpectrum.basicOpen (unop g))))
            (ModuleCat.Hom.hom
              ((Functor.obj modulesSpecToSheaf M).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf
                      (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
                (Quiver.Hom.op (homOfLE ⋯))))
            ⋯)) =
    ModuleCat.Hom.hom
      (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) (PrimeSpectrum.basicOpen (unop f)) ≫
        ModuleCat.ofHom
            (IsLocalizedModule.lift (Submonoid.powers (unop f))
              (ModuleCat.Hom.hom
                (tilde.toOpen
                  ((Functor.obj modulesSpecToSheaf M).obj (P :=
                        CategoryTheory.Presheaf.IsSheaf
                          (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
                    (op ⊤))
                  (PrimeSpectrum.basicOpen (unop f))))
              (ModuleCat.Hom.hom
                ((Functor.obj modulesSpecToSheaf M).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
                  (Quiver.Hom.op (homOfLE ⋯))))
              ⋯) ≫
          (Functor.obj modulesSpecToSheaf M).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
            i.unop.hom.op)

Tactic:
ext x