-- Module: Mathlib.AlgebraicGeometry.Modules.Sheaf
Goal:
X Y Z T : Scheme
M N K : X.Modules
φ : M ⟶ N
U V : X.Opens
f : X ⟶ Y
g : Y ⟶ Z
inst : IsOpenImmersion f
inst_1 : IsOpenImmersion g
this : (Hom.opensFunctor f ⋙ Opens.map f.base).IsContinuous (Opens.grothendieckTopology ↥X)
  (Opens.grothendieckTopology ↥X) :=
  ⋯
h :
  ∀ (U : (TopologicalSpace.Opens ↥X)ᵒᵖ)
    (x : (↑(X.ringCatSheaf.obj (P := Presheaf.IsSheaf (Opens.grothendieckTopology ↥X)).obj U) : Type u)),
    Eq (α :=
      (↑((ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf (Opens.grothendieckTopology ↥X))
                (Functor.obj
                  (Functor.sheafPushforwardContinuous (𝟭 X.Opens) RingCat (Opens.grothendieckTopology ↥X)
                    (Opens.grothendieckTopology ↥X))
                  X.ringCatSheaf)).obj
            U) :
        Type u))
      ((RingCat.Hom.hom
              (NatTrans.app
                (InducedCategory.Hom.hom
                  (CategoryStruct.comp (Y :=
                    Functor.obj
                      ((Hom.opensFunctor f ⋙ Opens.map f.base).sheafPushforwardContinuous RingCat
                        (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥X))
                      X.ringCatSheaf)
                    (CategoryStruct.comp (Y :=
                        Functor.obj
                          ((Hom.opensFunctor f).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
                            (Opens.grothendieckTopology ↥Y))
                          Y.ringCatSheaf)
                        {
                          hom :=
                            Functor.whiskerRight
                              {
                                app := fun (U : (TopologicalSpace.Opens ↥X)ᵒᵖ) ↦
                                  Iso.inv (Hom.appIso f (Opposite.unop U)),
                                naturality := ⋯ }
                              (forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                                (CC := CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦
                                (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)) }
                        (CategoryTheory.Functor.map
                          ((Hom.opensFunctor f).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
                            (Opens.grothendieckTopology ↥Y))
                          (Hom.toRingCatSheafHom f)) :
                      X.ringCatSheaf ⟶
                        Functor.obj
                          ((Hom.opensFunctor f ⋙ Opens.map f.base).sheafPushforwardContinuous RingCat
                            (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥X))
                          X.ringCatSheaf)
                    (NatTrans.app
                        (Functor.sheafPushforwardContinuousNatTrans
                          (Iso.hom (NatIso.ofComponents (fun (U : X.Opens) ↦ eqToIso ⋯) ⋯)) RingCat
                          (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥X))
                        X.ringCatSheaf :
                      Quiver.Hom (V := Sheaf (C := X.Opens) (Opens.grothendieckTopology ↥X) RingCat)
                        (Functor.obj
                          ((Hom.opensFunctor f ⋙ Opens.map f.base).sheafPushforwardContinuous RingCat
                            (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥X))
                          X.ringCatSheaf)
                        (Functor.obj
                          (Functor.sheafPushforwardContinuous (𝟭 X.Opens) RingCat (Opens.grothendieckTopology ↥X)
                            (Opens.grothendieckTopology ↥X))
                          X.ringCatSheaf))))
                U) :
            (a : (↑(X.ringCatSheaf.obj.obj U) : Type u)) →
              (↑((((𝟭 X.Opens).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
                              (Opens.grothendieckTopology ↥X)).obj
                          X.ringCatSheaf).obj.obj
                    U) :
                Type u))
          x :
        (↑((ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf (Opens.grothendieckTopology ↥X))
                  (Functor.obj
                    (Functor.sheafPushforwardContinuous (𝟭 X.Opens) RingCat (Opens.grothendieckTopology ↥X)
                      (Opens.grothendieckTopology ↥X))
                    X.ringCatSheaf)).obj
              U) :
          Type u))
      ((RingCat.Hom.hom ((InducedCategory.Hom.hom (𝟙 X.ringCatSheaf)).app U) :
            (a : (↑(X.ringCatSheaf.obj.obj U) : Type u)) →
              (↑((((𝟭 X.Opens).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
                              (Opens.grothendieckTopology ↥X)).obj
                          X.ringCatSheaf).obj.obj
                    U) :
                Type u))
          x :
        (↑((ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf (Opens.grothendieckTopology ↥X))
                  (Functor.obj
                    (Functor.sheafPushforwardContinuous (𝟭 X.Opens) RingCat (Opens.grothendieckTopology ↥X)
                      (Opens.grothendieckTopology ↥X))
                    X.ringCatSheaf)).obj
              U) :
          Type u))
⊢ CategoryStruct.comp (Y :=
      Functor.obj
        ((Hom.opensFunctor f ⋙ Opens.map f.base).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
          (Opens.grothendieckTopology ↥X))
        X.ringCatSheaf)
      (CategoryStruct.comp (Y :=
          Functor.obj
            ((Hom.opensFunctor f).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
              (Opens.grothendieckTopology ↥Y))
            Y.ringCatSheaf)
          {
            hom :=
              Functor.whiskerRight
                { app := fun (U : (TopologicalSpace.Opens ↥X)ᵒᵖ) ↦ Iso.inv (Hom.appIso f (Opposite.unop U)),
                  naturality := ⋯ }
                (forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
                  RingCat.carrier)) }
          (CategoryTheory.Functor.map
            ((Hom.opensFunctor f).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
              (Opens.grothendieckTopology ↥Y))
            (Hom.toRingCatSheafHom f)) :
        X.ringCatSheaf ⟶
          Functor.obj
            ((Hom.opensFunctor f ⋙ Opens.map f.base).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
              (Opens.grothendieckTopology ↥X))
            X.ringCatSheaf)
      (NatTrans.app
          (Functor.sheafPushforwardContinuousNatTrans (Iso.hom (NatIso.ofComponents (fun (U : X.Opens) ↦ eqToIso ⋯) ⋯))
            RingCat (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥X))
          X.ringCatSheaf :
        Quiver.Hom (V := Sheaf (C := X.Opens) (Opens.grothendieckTopology ↥X) RingCat)
          (Functor.obj
            ((Hom.opensFunctor f ⋙ Opens.map f.base).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
              (Opens.grothendieckTopology ↥X))
            X.ringCatSheaf)
          (Functor.obj
            (Functor.sheafPushforwardContinuous (𝟭 X.Opens) RingCat (Opens.grothendieckTopology ↥X)
              (Opens.grothendieckTopology ↥X))
            X.ringCatSheaf)) =
    𝟙 X.ringCatSheaf

Tactic:
ext U x