Module: Mathlib.AlgebraicGeometry.Modules.Sheaf

Goal state before:
pp: X✝ Y Z T : Scheme
M N K : X✝.Modules
φ : M ⟶ N
U✝ V✝ : X✝.Opens
X : Scheme
U V : X.Opens
f : V ⟶ U
this✝⁴ :
  (Hom.opensFunctor (X.homOfLE ⋯)).IsContinuous (Opens.grothendieckTopology ↥↑V) (Opens.grothendieckTopology ↑U.carrier)
this✝³ :
  Functor.IsDenseSubsite (Opens.grothendieckTopology ↥U) ((Opens.grothendieckTopology ↥X).over U)
    (Opens.overEquivalence U).inverse
this✝² : (Hom.opensFunctor (X.homOfLE ⋯)).IsContinuous (Opens.grothendieckTopology ↥V) (Opens.grothendieckTopology ↥↑U)
this✝¹ :
  ((Opens.overEquivalence V).symm.functor ⋙ Over.map f).IsContinuous (Opens.grothendieckTopology ↥V)
    ((Opens.grothendieckTopology ↥X).over U)
this✝ :
  (Opens.overEquivalence U).symm.functor.IsContinuous (Opens.grothendieckTopology ↥↑U)
    ((Opens.grothendieckTopology ↥X).over U)
this :
  (Hom.opensFunctor (X.homOfLE ⋯) ⋙ (Opens.overEquivalence U).symm.functor).IsContinuous (Opens.grothendieckTopology ↥V)
    ((Opens.grothendieckTopology ↥X).over U)
⊢ SheafOfModules.pushforward
      (((Opens.overPullbackSheafEquivOver V).app X.ringCatSheaf).inv ≫
        ((Opens.overEquivalence V).symm.functor.sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥V)
              ((Opens.grothendieckTopology ↥X).over V)).map
          (Sheaf.pushforwardOverMapIso X.ringCatSheaf f).inv) ≅
    SheafOfModules.pushforward
      ({
          hom :=
            Functor.whiskerRight
              { app := fun U_1 ↦ (Hom.appIso (X.homOfLE ⋯) (Opposite.unop U_1)).inv, naturality := ⋯ }
              (forget₂ CommRingCat RingCat) } ≫
        ((Hom.opensFunctor (X.homOfLE ⋯)).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥↑V)
              (Opens.grothendieckTopology ↥U)).map
          ((Opens.overPullbackSheafEquivOver U).app X.ringCatSheaf).inv)

Tactic:
refine SheafOfModules.pushforwardCongr₂ _ ?_ ?_

Goal state after:
case refine_1
X✝ Y Z T : Scheme
M N K : X✝.Modules
φ : M ⟶ N
U✝ V✝ : X✝.Opens
X : Scheme
U V : X.Opens
f : V ⟶ U
this✝⁴ :
  (Hom.opensFunctor (X.homOfLE ⋯)).IsContinuous (Opens.grothendieckTopology ↥↑V) (Opens.grothendieckTopology ↑U.carrier)
this✝³ :
  Functor.IsDenseSubsite (Opens.grothendieckTopology ↥U) ((Opens.grothendieckTopology ↥X).over U)
    (Opens.overEquivalence U).inverse
this✝² : (Hom.opensFunctor (X.homOfLE ⋯)).IsContinuous (Opens.grothendieckTopology ↥V) (Opens.grothendieckTopology ↥↑U)
this✝¹ :
  ((Opens.overEquivalence V).symm.functor ⋙ Over.map f).IsContinuous (Opens.grothendieckTopology ↥V)
    ((Opens.grothendieckTopology ↥X).over U)
this✝ :
  (Opens.overEquivalence U).symm.functor.IsContinuous (Opens.grothendieckTopology ↥↑U)
    ((Opens.grothendieckTopology ↥X).over U)
this :
  (Hom.opensFunctor (X.homOfLE ⋯) ⋙ (Opens.overEquivalence U).symm.functor).IsContinuous (Opens.grothendieckTopology ↥V)
    ((Opens.grothendieckTopology ↥X).over U)
⊢ Hom.opensFunctor (X.homOfLE ⋯) ⋙ (Opens.overEquivalence U).symm.functor ≅
    (Opens.overEquivalence V).symm.functor ⋙ Over.map f

case refine_2
X✝ Y Z T : Scheme
M N K : X✝.Modules
φ : M ⟶ N
U✝ V✝ : X✝.Opens
X : Scheme
U V : X.Opens
f : V ⟶ U
this✝⁴ :
  (Hom.opensFunctor (X.homOfLE ⋯)).IsContinuous (Opens.grothendieckTopology ↥↑V) (Opens.grothendieckTopology ↑U.carrier)
this✝³ :
  Functor.IsDenseSubsite (Opens.grothendieckTopology ↥U) ((Opens.grothendieckTopology ↥X).over U)
    (Opens.overEquivalence U).inverse
this✝² : (Hom.opensFunctor (X.homOfLE ⋯)).IsContinuous (Opens.grothendieckTopology ↥V) (Opens.grothendieckTopology ↥↑U)
this✝¹ :
  ((Opens.overEquivalence V).symm.functor ⋙ Over.map f).IsContinuous (Opens.grothendieckTopology ↥V)
    ((Opens.grothendieckTopology ↥X).over U)
this✝ :
  (Opens.overEquivalence U).symm.functor.IsContinuous (Opens.grothendieckTopology ↥↑U)
    ((Opens.grothendieckTopology ↥X).over U)
this :
  (Hom.opensFunctor (X.homOfLE ⋯) ⋙ (Opens.overEquivalence U).symm.functor).IsContinuous (Opens.grothendieckTopology ↥V)
    ((Opens.grothendieckTopology ↥X).over U)
⊢ (((Opens.overPullbackSheafEquivOver V).app X.ringCatSheaf).inv ≫
        ((Opens.overEquivalence V).symm.functor.sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥V)
              ((Opens.grothendieckTopology ↥X).over V)).map
          (Sheaf.pushforwardOverMapIso X.ringCatSheaf f).inv) ≫
      (Functor.sheafPushforwardContinuousNatTrans (Iso.hom ?refine_1) RingCat (Opens.grothendieckTopology ↥V)
            ((Opens.grothendieckTopology ↥X).over U)).app
        (Sheaf.over X.ringCatSheaf U) =
    {
        hom :=
          Functor.whiskerRight { app := fun U_1 ↦ (Hom.appIso (X.homOfLE ⋯) (Opposite.unop U_1)).inv, naturality := ⋯ }
            (forget₂ CommRingCat RingCat) } ≫
      ((Hom.opensFunctor (X.homOfLE ⋯)).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥↑V)
            (Opens.grothendieckTopology ↥U)).map
        ((Opens.overPullbackSheafEquivOver U).app X.ringCatSheaf).inv