-- 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 ⋙ Hom.opensFunctor g).IsContinuous (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥Z)
h :
  ∀ (x : (TopologicalSpace.Opens ↥X)ᵒᵖ),
    Eq (α :=
      X.ringCatSheaf.obj (P := Presheaf.IsSheaf (Opens.grothendieckTopology ↥X)).obj x ⟶
        (ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf (Opens.grothendieckTopology ↥X))
              (Functor.obj
                ((Hom.opensFunctor f ⋙ Hom.opensFunctor g).sheafPushforwardContinuous RingCat
                  (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥Z))
                Z.ringCatSheaf)).obj
          x)
      (NatTrans.app
        (InducedCategory.Hom.hom
          (CategoryStruct.comp (Y :=
            Functor.obj
              ((Hom.opensFunctor (f ≫ g)).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
                (Opens.grothendieckTopology ↥Z))
              Z.ringCatSheaf)
            {
              hom :=
                Functor.whiskerRight
                  { app := fun (U : (TopologicalSpace.Opens ↥X)ᵒᵖ) ↦ Iso.inv (Hom.appIso (f ≫ g) (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)) }
            (NatTrans.app
              (Functor.sheafPushforwardContinuousNatTrans
                (Iso.hom (NatIso.ofComponents (fun (x : X.Opens) ↦ eqToIso ⋯) ⋯)) RingCat
                (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥Z))
              Z.ringCatSheaf)))
        x)
      (NatTrans.app
        (InducedCategory.Hom.hom
          (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 :=
                  Functor.whiskerRight
                    { app := fun (U : (TopologicalSpace.Opens ↥Y)ᵒᵖ) ↦ Iso.inv (Hom.appIso g (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)) })))
        x)
⊢ Eq (α :=
    X.ringCatSheaf ⟶
      Functor.obj
        ((Hom.opensFunctor f ⋙ Hom.opensFunctor g).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
          (Opens.grothendieckTopology ↥Z))
        Z.ringCatSheaf)
    (CategoryStruct.comp (Y :=
      Functor.obj
        ((Hom.opensFunctor (f ≫ g)).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
          (Opens.grothendieckTopology ↥Z))
        Z.ringCatSheaf)
      {
        hom :=
          Functor.whiskerRight
            { app := fun (U : (TopologicalSpace.Opens ↥X)ᵒᵖ) ↦ Iso.inv (Hom.appIso (f ≫ g) (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)) }
      (NatTrans.app
        (Functor.sheafPushforwardContinuousNatTrans (Iso.hom (NatIso.ofComponents (fun (x : X.Opens) ↦ eqToIso ⋯) ⋯))
          RingCat (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥Z))
        Z.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 :=
            Functor.whiskerRight
              { app := fun (U : (TopologicalSpace.Opens ↥Y)ᵒᵖ) ↦ Iso.inv (Hom.appIso g (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)) }))

Tactic:
ext : 3