-- Module: Mathlib.CategoryTheory.Galois.GaloisObjects
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
A B : C
f : A ⟶ B
inst_2 : IsConnected.{u₁, u₂} A
inst_3 : IsGalois.{u₁, u₂} B
σ : Aut A
F : C ⥤ FintypeCat := GaloisCategory.getFiberFunctor C
a : (F.obj A).obj
refine_1 : Aut B
refine_2 :
  (fun (τ : Aut B) ↦ f ≫ τ.hom = σ.hom ≫ f)
    (((evaluationEquivOfIsGalois F 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)).symm :
        (a : (F.obj B).obj) → Aut 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 ≫ 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)))
refine_3 :
  ∀ (y : Aut B),
    (fun (τ : Aut B) ↦ f ≫ τ.hom = σ.hom ≫ f) y →
      y =
        ((evaluationEquivOfIsGalois F 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)).symm :
            (a : (F.obj B).obj) → Aut 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 ≫ 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))
⊢ ∃! τ : Aut B, f ≫ τ.hom = σ.hom ≫ f

Tactic:
refine ⟨?_, ?_, ?_⟩