∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor C CategoryTheory.Cat}
  {E : Type u_1} [inst_1 : CategoryTheory.Category.{v_1, u_1} E] (fib : (c : C) → CategoryTheory.Functor (↑(F.obj c)) E)
  (hom : {c c' : C} → (f : c ⟶ c') → fib c ⟶ (F.map f).toFunctor.comp (fib c'))
  (hom_id : ∀ (c : C), hom (CategoryTheory.CategoryStruct.id c) = CategoryTheory.eqToHom ⋯)
  (hom_comp :
    ∀ (c₁ c₂ c₃ : C) (f : c₁ ⟶ c₂) (g : c₂ ⟶ c₃),
      hom (CategoryTheory.CategoryStruct.comp f g) =
        CategoryTheory.CategoryStruct.comp (hom f)
          (CategoryTheory.CategoryStruct.comp ((F.map f).toFunctor.whiskerLeft (hom g)) (CategoryTheory.eqToHom ⋯)))
  (X : CategoryTheory.Grothendieck F),
  (CategoryTheory.Grothendieck.functorFrom fib hom hom_id hom_comp).obj X = (fib X.base).obj X.fiber