-- Module: Mathlib.CategoryTheory.Galois.Topology
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_1 : GaloisCategory.{u₁, u₂} C
inst_2 : FiberFunctor F
U : Set ((X : C) → Aut (F.obj X))
hUopen : IsOpen U
h1 : Membership.mem (γ := Set (Aut F)) ((⇑(autEmbedding F) : (a : Aut F) → (X : C) → Aut (F.obj X)) ⁻¹' U) 1
h_1 : IsOpen (X := Aut F) ((⇑(autEmbedding F) : (a : Aut F) → (X : C) → Aut (F.obj X)) ⁻¹' U)
I : Finset C
u : (a : C) → Set (Aut (F.obj a))
ho : ∀ a ∈ I, IsOpen (u a) ∧ 1 a ∈ u a
ha : (↑I : Set C).pi (α := fun (a : C) ↦ Aut (F.obj a)) u ⊆ U
fι : (X : ↥I) → Type
ff : (X : ↥I) → fι X → C
fc : (X : ↥I) → (i : fι X) → ff X i ⟶ (↑X : C)
h4 : (X : ↥I) → Limits.IsColimit (Limits.Cofan.mk (↑X : C) (fc X))
h5 : ∀ (X : ↥I) (i : fι X), IsConnected.{u₁, u₂} (ff X i)
h6 : ∀ (X : ↥I), Finite (fι X)
σ : Aut F
h : ∀ (X : (↑(⋃ (X : ↥I), Set.range (ff X)) : Type u₁)), σ.hom.app (↑X : C) = 𝟙 (F.obj (↑X : C))
X : C
XinI : X ∈ (↑I : Set C)
this : Fintype (fι (Subtype.mk (p := fun (x : C) ↦ x ∈ I) X XinI))
x : (F.obj X).obj
h₁ :
  ∀ (j : fι (Subtype.mk (p := fun (x : C) ↦ x ∈ I) X XinI))
    (a :
      (obj (Discrete.functor (ff (Subtype.mk (p := fun (x : C) ↦ x ∈ I) X XinI)) ⋙ F)
          { as := j : Discrete (fι (Subtype.mk (p := fun (x : C) ↦ x ∈ I) X XinI)) }).obj)
    (ha :
      (@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.map
                  ((Limits.Cofan.mk (↑(Subtype.mk (p := fun (x : C) ↦ x ∈ I) X XinI) : C)
                          (fc (Subtype.mk (p := fun (x : C) ↦ x ∈ I) X XinI))).ι.app
                    { as := j }) :
                F.obj
                    (obj.{0, u₂, 0, u₁} (Discrete.functor (ff (Subtype.mk (p := fun (x : C) ↦ x ∈ I) X XinI)))
                      { as := j : Discrete (fι (Subtype.mk (p := fun (x : C) ↦ x ∈ I) X XinI)) }) ⟶
                  F.obj X) :
            (a :
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                  (F.obj ((Discrete.functor (ff ⟨X, XinI⟩)).obj { as := j }))) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
          a =
        x),
    Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
      ((@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))
              _ _ _ (σ.hom.app X) :
            (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
          x :
        (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
      ((@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))
              _ _ _ (𝟙 (F.obj X) : F.obj X ⟶ F.obj X) :
            (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
          x :
        (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
⊢ Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
    ((@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))
            _ _ _ (σ.hom.app X) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        x :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
    ((@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))
            _ _ _ (𝟙 (F.obj X) : F.obj X ⟶ F.obj X) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        x :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))

Tactic:
obtain ⟨⟨j⟩, a, ha : F.map _ a = x⟩ := Limits.FintypeCat.jointly_surjective
        (Discrete.functor (ff ⟨X, XinI⟩) ⋙ F) _ (Limits.isColimitOfPreserves F (h4 ⟨X, XinI⟩)) x

Premises used: CategoryTheory.Functor.comp, CategoryTheory.Discrete.functor, Subtype.mk, CategoryTheory.Limits.FintypeCat.jointly_surjective, CategoryTheory.Limits.isColimitOfPreserves