-- Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
eqv : CategoryTheory.Equivalence.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : eqv.functor.IsContinuous J K
inst_3 : eqv.inverse.IsContinuous K J
φ : S ⟶ obj (eqv.functor.sheafPushforwardContinuous RingCat J K) R
ψ : R ⟶ obj (eqv.inverse.sheafPushforwardContinuous RingCat K J) S
H₁ : whiskerRight (NatTrans.op eqv.counit) (R.obj (P := Presheaf.IsSheaf K)) = ψ.hom ≫ eqv.inverse.op.whiskerLeft φ.hom
H₂ :
  φ.hom ≫ eqv.functor.op.whiskerLeft ψ.hom ≫ whiskerRight (NatTrans.op eqv.unit) (S.obj (P := Presheaf.IsSheaf J)) =
    𝟙 (S.obj (P := Presheaf.IsSheaf J))
M : SheafOfModules.{u_1, v₂, u₂, u} R
U : Dᵒᵖ
x : (↑((obj (𝟭 (SheafOfModules.{u_1, v₂, u₂, u} R)) M).val.obj U) : Type u_1)
⊢ Eq (α :=
    (↑((((pushforwardPushforwardEquivalence eqv φ ψ H₁ H₂).functor ⋙
                    (pushforwardPushforwardEquivalence eqv φ ψ H₁ H₂).inverse).obj
                M).val.obj
          U) :
      Type u_1))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat.{u_1, u} (↑(R.obj (P := Presheaf.IsSheaf K).obj U) : Type u)) ↦
            LinearMap.{u, u, u_1, u_1} (R := (↑(R.obj (P := Presheaf.IsSheaf K).obj U) : Type u))
              (RingHom.id (↑(R.obj (P := Presheaf.IsSheaf K).obj U) : Type u)) (↑x1 : Type u_1) (↑x2 : Type u_1))
            (CC := ModuleCat.carrier) (((pushforwardPushforwardEquivalence eqv φ ψ H₁ H₂).unit.app M).val.app U) :
          (a : (↑(((𝟭 (SheafOfModules R)).obj M).val.obj U) : Type u_1)) →
            (↑((((pushforwardPushforwardEquivalence eqv φ ψ H₁ H₂).functor ⋙
                            (pushforwardPushforwardEquivalence eqv φ ψ H₁ H₂).inverse).obj
                        M).val.obj
                  U) :
              Type u_1))
        x :
      (↑((((pushforwardPushforwardEquivalence eqv φ ψ H₁ H₂).functor ⋙
                      (pushforwardPushforwardEquivalence eqv φ ψ H₁ H₂).inverse).obj
                  M).val.obj
            U) :
        Type u_1))
    ((ConcreteCategory.hom (FC :=
            fun
              (x1 x2 :
                ModuleCat.{u_1, u}
                  (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (obj.{v₂, v₂, u₂, u₂} (𝟭 D) (Opposite.unop U)))) :
                    Type u)) ↦
            LinearMap.{u, u, u_1, u_1} (R :=
              (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (obj.{v₂, v₂, u₂, u₂} (𝟭 D) (Opposite.unop U)))) :
                Type u))
              (RingHom.id
                (↑(R.obj (P := Presheaf.IsSheaf K).obj (Opposite.op (obj.{v₂, v₂, u₂, u₂} (𝟭 D) (Opposite.unop U)))) :
                  Type u))
              (↑x1 : Type u_1) (↑x2 : Type u_1))
            (CC := ModuleCat.carrier) (Y :=
            obj
              (ModuleCat.restrictScalars
                (RingCat.Hom.hom
                  (R.obj (P := Presheaf.IsSheaf K).map (Quiver.Hom.op (eqv.counit.app (Opposite.unop U))))))
              (M.val.obj (Opposite.op ((eqv.inverse ⋙ eqv.functor).obj (Opposite.unop U)))))
            (M.val.map (Quiver.Hom.op (eqv.counit.app (Opposite.unop U)))) :
          (a : (↑(M.val.obj (Opposite.op ((𝟭 D).obj (Opposite.unop U)))) : Type u_1)) →
            (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (R.obj.map (eqv.counit.app (Opposite.unop U)).op))).obj
                  (M.val.obj (Opposite.op ((eqv.inverse ⋙ eqv.functor).obj (Opposite.unop U))))) :
              Type u_1))
        x :
      (↑(obj
            (ModuleCat.restrictScalars
              (RingCat.Hom.hom
                (R.obj (P := Presheaf.IsSheaf K).map (Quiver.Hom.op (eqv.counit.app (Opposite.unop U))))))
            (M.val.obj (Opposite.op ((eqv.inverse ⋙ eqv.functor).obj (Opposite.unop U))))) :
        Type u_1))

Tactic:
rfl

Premises used: rfl