-- Module: Mathlib.CategoryTheory.Galois.EssSurj
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, u₁, u₁, u₁ + 1} C FintypeCat
inst_1 : GaloisCategory.{u₁, u₂} C
inst_2 : FiberFunctor F
G : Type u_1
inst_3 : Group G
inst_4 : TopologicalSpace G
inst_5 : IsTopologicalGroup G
inst_6 : CompactSpace G
V U : OpenSubgroup (Aut F)
h : (↑U : Subgroup (Aut F)).Normal
A : C
u : (functorToAction F).obj A ≅ ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ (↑U : Subgroup (Aut F))))
hUinV : U ≤ V
s : Cocone (quotientDiag V h u ⋙ functorToAction F)
j :
  SingleObj.{u₁}
    (HasQuotient.Quotient ↥(↑V : Subgroup (Aut F)) (B := Subgroup ↥(↑V : Subgroup (Aut F)))
      ((↑U : Subgroup (Aut F)).subgroupOf (↑V : Subgroup (Aut F))))
x : Aut F ⧸ (↑U : Subgroup (Aut F))
h₁ :
  ∀ (a : Aut F),
    Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
      ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
              _ _ _
              (Action.Hom.hom
                  (u.inv ≫
                    NatTrans.app (Cocone.ι (coconeQuotientDiag h u hUinV)) j ≫ coconeQuotientDiagDesc h u hUinV s) :
                Action.V (G := Aut F) (ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ (↑U : Subgroup (Aut F))))) ⟶
                  s.pt.V) :
            (a :
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                  (ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ (↑U : Subgroup (Aut F))))).V) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
          ⟦a⟧ :
        (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
      ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
              _ _ _ (u.inv ≫ s.ι.app j).hom :
            (a :
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                  (ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ (↑U : Subgroup (Aut F))))).V) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
          ⟦a⟧ :
        (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
⊢ Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _
            (Action.Hom.hom
                (u.inv ≫
                  NatTrans.app (Cocone.ι (coconeQuotientDiag h u hUinV)) j ≫ coconeQuotientDiagDesc h u hUinV s) :
              Action.V (G := Aut F) (ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ (↑U : Subgroup (Aut F))))) ⟶ s.pt.V) :
          (a :
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                (ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ (↑U : Subgroup (Aut F))))).V) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
        x :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (u.inv ≫ s.ι.app j).hom :
          (a :
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                (ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ (↑U : Subgroup (Aut F))))).V) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
        x :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)

Tactic:
induction x using Quotient.inductionOn