-- 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_2 :
  Eq.{u₂ + 1} (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B))
    ((fun (f : A ⟶ 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)))
      (f ≫
        Iso.hom
          (((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)))))
    ((fun (f : A ⟶ 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)))
      (σ.hom ≫ f))
⊢ (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)))

Tactic:
apply evaluation_injective_of_isConnected F A B a

Premises used: CategoryTheory.PreGaloisCategory.evaluation_injective_of_isConnected