-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
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 ⊤)
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))
have hf : 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))
  ((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 sorry))))
          sorry :
        (a :
            (↑((modulesSpecToSheaf.obj (tilde ((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)))).presheaf.obj
                  (op (PrimeSpectrum.basicOpen (unop g)))) :
              Type u)) →
          (↑((modulesSpecToSheaf.obj M).obj.obj (op (PrimeSpectrum.basicOpen (unop g)))) : Type u))
      ((ModuleCat.Hom.hom
              (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤))
                (PrimeSpectrum.basicOpen (unop g))) :
            (a : (↑((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)) : Type u)) →
              (↑((modulesSpecToSheaf.obj (tilde ((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)))).presheaf.obj
                    (op (PrimeSpectrum.basicOpen (unop g)))) :
                Type u))
          x :
        (↑((Functor.obj modulesSpecToSheaf
                    (tilde ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)))).presheaf.obj
              (op (PrimeSpectrum.basicOpen (unop g)))) :
          Type u)) :
    (↑((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
          ((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).obj.obj (op (PrimeSpectrum.basicOpen (unop f)))) : Type u)) →
          (↑((modulesSpecToSheaf.obj M).obj.obj (op (PrimeSpectrum.basicOpen (unop g)))) : Type u))
      ((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 sorry))))
              sorry :
            (a :
                (↑((modulesSpecToSheaf.obj (tilde ((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)))).obj.obj
                      (op (PrimeSpectrum.basicOpen (unop f)))) :
                  Type u)) →
              (↑((modulesSpecToSheaf.obj M).obj.obj (op (PrimeSpectrum.basicOpen (unop f)))) : Type u))
          ((ModuleCat.Hom.hom
                  (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤))
                    (PrimeSpectrum.basicOpen (unop f))) :
                (a : (↑((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)) : Type u)) →
                  (↑((modulesSpecToSheaf.obj (tilde ((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)))).obj.obj
                        (op (PrimeSpectrum.basicOpen (unop f)))) :
                    Type u))
              x :
            (↑((Functor.obj modulesSpecToSheaf (tilde ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)))).obj
                      (P :=
                      CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
                  (op (PrimeSpectrum.basicOpen (unop f)))) :
              Type u)) :
        (↑((Functor.obj modulesSpecToSheaf M).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf
                    (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
              (op (PrimeSpectrum.basicOpen (unop f)))) :
          Type u)) :
    (↑((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))
Tactic: dsimp