-- 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)
f : Cocone.pt (F := quotientDiag V h u ⋙ functorToAction F) (coconeQuotientDiag h u hUinV) ⟶ s.pt
hf :
  ∀
    (j :
      SingleObj.{u₁}
        (HasQuotient.Quotient ↥(↑V : Subgroup (Aut F)) (B := Subgroup ↥(↑V : Subgroup (Aut F)))
          ((↑U : Subgroup (Aut F)).subgroupOf (↑V : Subgroup (Aut F))))),
    NatTrans.app (Cocone.ι (coconeQuotientDiag h u hUinV)) j ≫ f = s.ι.app j
x : Aut F ⧸ (↑V : Subgroup (Aut F))
h₁ :
  ∀ (σ : 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))
              _ _ _ f.hom :
            (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (coconeQuotientDiag h u hUinV).pt.V) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
          ⟦σ⟧ :
        (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))
              _ _ _ (coconeQuotientDiagDesc h u hUinV s).hom :
            (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (coconeQuotientDiag h u hUinV).pt.V) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) s.pt.V)
          ⟦σ⟧ :
        (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))
            _ _ _ f.hom :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (coconeQuotientDiag h u hUinV).pt.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))
            _ _ _ (coconeQuotientDiagDesc h u hUinV s).hom :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (coconeQuotientDiag h u hUinV).pt.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 with | _ σ