-- Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree
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
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ Functor.obj (F.sheafPushforwardContinuous RingCat J K) R
inst_3 :
  J.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_4 :
  K.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_5 : (pushforward φ).IsRightAdjoint
inst_6 : F.Final
M : SheafOfModules R
f : unit R ⟶ M
h :
  Eq (α := ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
            (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
          (pullbackObjUnitToUnit φ ≫ f)) :
      ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f) :
      ((pushforward φ).obj M).sections)
h₁ :
  Eq (α := unit S ⟶ (pushforward φ).obj M)
    ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
          (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
        (pullbackObjUnitToUnit φ ≫ f) :
      unit S ⟶ (pushforward φ).obj M)
    (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f : unit S ⟶ (pushforward φ).obj M)
h₂ :
  Eq (α := unit S ⟶ (pushforward φ).obj M)
    ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) (unit R) :
            (a : (pullback φ).obj (unit S) ⟶ unit R) → unit S ⟶ (pushforward φ).obj (unit R))
          (pullbackObjUnitToUnit φ) ≫
        (pushforward φ).map f :
      unit S ⟶ (pushforward φ).obj M)
    (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f : unit S ⟶ (pushforward φ).obj M)
h₃ h₄ :
  Eq (α := unit S ⟶ (pushforward φ).obj M) (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f)
    (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f)
⊢ Eq (α := ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
            (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
          (pullbackObjUnitToUnit φ ≫ f)) :
      ((pushforward φ).obj M).sections)
    (pushforwardSections φ ((M.unitHomEquiv : (a : unit R ⟶ M) → M.sections) f) : ((pushforward φ).obj M).sections)

Tactic:
rw [pushforwardSections_unitHomEquiv, EmbeddingLike.apply_eq_iff_eq,
    Adjunction.homEquiv_naturality_right,
    pullbackPushforwardAdjunction_homEquiv_pullbackObjUnitToUnit]

Premises used: SheafOfModules.pullbackPushforwardAdjunction_homEquiv_pullbackObjUnitToUnit, EmbeddingLike.apply_eq_iff_eq, SheafOfModules.pushforwardSections_unitHomEquiv, CategoryTheory.Adjunction.homEquiv_naturality_right