-- 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))))
a : Aut F
h₁ :
  Eq (α := s.pt.V.obj)
    (((@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
                  (s.ι.app
                    (SingleObj.star
                      (HasQuotient.Quotient ↥(↑V : Subgroup (Aut F)) (B := Subgroup ↥(↑V : Subgroup (Aut F)))
                        ((↑U : Subgroup (Aut F)).subgroupOf (↑V : Subgroup (Aut F)))))) :
                ((functorToAction F).obj A).V ⟶ s.pt.V) :
            TypeCat.Fun ((functorToAction F).obj A).V.obj s.pt.V.obj) :
          (a : ((functorToAction F).obj A).V.obj) → s.pt.V.obj)
        ((@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.hom :
              (a : (ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ (↑U : Subgroup (Aut F))))).V.obj) →
                ((functorToAction F).obj A).V.obj)
            ⟦a⟧ :
          ((functorToAction F).obj A).V.obj) :
      s.pt.V.obj)
    ((@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))
            _ _ _ (s.ι.app j).hom :
          (a : ((functorToAction F).obj A).V.obj) → s.pt.V.obj)
        ((@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.hom :
              (a : (ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ (↑U : Subgroup (Aut F))))).V.obj) →
                ((functorToAction F).obj A).V.obj)
            ⟦a⟧ :
          ((functorToAction F).obj A).V.obj) :
      s.pt.V.obj)
⊢ 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)

Tactic:
simp