∀ {C : Type u} [inst : CategoryTheory.Groupoid C] {D : Type u_1} [inst_1 : CategoryTheory.Groupoid D]
  (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) {c d : D} (f : c ⟶ d),
  f ∈ (CategoryTheory.Subgroupoid.im φ hφ).arrows c d ↔
    ∃ a b g,
      ∃ (ha : φ.obj a = c) (hb : φ.obj b = d),
        f =
          CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ⋯)
            (CategoryTheory.CategoryStruct.comp (φ.map g) (CategoryTheory.eqToHom hb))