-- Module: Mathlib.CategoryTheory.Galois.GaloisObjects
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
A B : C
inst_2 : IsConnected.{u₁, u₂} A
inst_3 : IsGalois.{u₁, u₂} B
f : A ⟶ B
σ τ : Aut A
F : C ⥤ FintypeCat := GaloisCategory.getFiberFunctor C
a : (F.obj A).obj
a₁ :
  Eq.{u₂ + 1} (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B))
    ((fun (f_1 : 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 f_1.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)))
      (autMap f (σ * τ)))
    ((fun (f_1 : 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 f_1.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)))
      (autMap f σ * autMap f τ))
⊢ autMap f (σ * τ) = autMap f σ * autMap f τ

Tactic:
apply evaluation_aut_injective_of_isConnected F (B : C) (F.map f a)

Premises used: CategoryTheory.ConcreteCategory.hom, CategoryTheory.Functor.map, DFunLike.coe, CategoryTheory.PreGaloisCategory.evaluation_aut_injective_of_isConnected