-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.Prorepresentability (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : C ⥤ FintypeCat
inst_2 : FiberFunctor F
f g : End F
A : PointedGaloisObject F
⊢ Eq (α := (F.obj A.obj).obj)
    ((@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))
            _ _ _ (g.app A.obj ≫ f.app A.obj : F.obj A.obj ⟶ F.obj A.obj) :
          (a : (F.obj A.obj).obj) → (F.obj A.obj).obj)
        A.pt :
      (F.obj A.obj).obj)
    ((@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
                (CategoryStruct.comp (Y := A.obj)
                  (Iso.hom
                    ((AutGalois.π F A : (a : AutGalois F) → Aut A.obj)
                      ((endEquivAutGalois F : (a : End F) → AutGalois F) f)))
                  (Iso.hom
                    ((AutGalois.π F A : (a : AutGalois F) → Aut A.obj)
                      ((endEquivAutGalois F : (a : End F) → AutGalois F) g)))) :
              F.obj A.obj ⟶ F.obj A.obj) :
          (a : (F.obj A.obj).obj) → (F.obj A.obj).obj)
        A.pt :
      (F.obj A.obj).obj)
have simp : Eq (α := (F.obj A.obj).obj)
  ((@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.app A.obj : F.obj A.obj ⟶ F.obj A.obj) :
        (a : (F.obj A.obj).obj) → (F.obj A.obj).obj)
      ((@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))
              _ _ _ (g.app A.obj : F.obj A.obj ⟶ F.obj A.obj) :
            (a : (F.obj A.obj).obj) → (F.obj A.obj).obj)
          A.pt :
        (F.obj A.obj).obj) :
    (F.obj A.obj).obj)
  ((@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
              (Iso.hom
                ((AutGalois.π F A : (a : AutGalois F) → Aut A.obj)
                  ((endEquivAutGalois F : (a : End F) → AutGalois F) g))) :
            F.obj A.obj ⟶ F.obj A.obj) :
        (a : (F.obj A.obj).obj) → (F.obj A.obj).obj)
      ((@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.app A.obj : F.obj A.obj ⟶ F.obj A.obj) :
            (a : (F.obj A.obj).obj) → (F.obj A.obj).obj)
          A.pt :
        (F.obj A.obj).obj) :
    (F.obj A.obj).obj)
Tactic: simp only [map_comp, FintypeCat.comp_apply, endEquivAutGalois_π]