-- 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)
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)

Tactic:
simp [← Functor.map_comp, SheafedSpace.sheaf]

Premises used: CategoryTheory.Functor.map_comp, AlgebraicGeometry.SheafedSpace.sheaf