-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.GaloisObjects (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
A B : C
inst_2 : IsGalois.{u₁, u₂} A
inst_3 : IsGalois.{u₁, u₂} B
f : A ⟶ B
σ : Aut B
F : C ⥤ FintypeCat := GaloisCategory.getFiberFunctor C
a a' : (F.obj A).obj
ha' :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B))
    ((@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 f) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B))
        a' :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B))
    ((@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 σ.hom) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B))
        ((@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 f) :
              (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B))
            a :
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B)) :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B))
⊢ ∃ (a : Aut A), autMap f a = σ
have obtain : ∀ (τ : Aut A)
  (hτ :
    (@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 τ.hom) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A))
        a =
      a'),
  ∃ (a : Aut A), autMap f a = σ
Tactic: obtain ⟨τ, (hτ : F.map τ.hom a = a')⟩ := MulAction.exists_smul_eq (Aut A) a a'