-- Module: Mathlib.AlgebraicGeometry.Modules.Sheaf
Goal:
X Y Z T : Scheme
M N K : X.Modules
φ : M ⟶ N
U_1 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) :=
  ⋯
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))

Tactic:
exact
      congr($(f.appIso_inv_app_presheafMap U.unop) x)