∀ {k : Type u_4} {G : Type u_3} {H : Type u_2} [inst : CommRing k] [inst_1 : Monoid G] [inst_2 : Monoid H] (φ : G →* H)
  (B : Rep.{max u_2 u_1, u_4, u_2} k H) (A : Rep.{max u_2 u_1, u_4, u_3} k G) (x : Rep.res φ B ⟶ A),
  (fun f =>
        Rep.ofHom
          { toLinearMap := LinearMap.proj 1 ∘ₗ (Representation.coindV φ A.ρ).subtype ∘ₗ (Rep.Hom.hom f).toLinearMap,
            isIntertwining' := ⋯ })
      ((fun f => Rep.resCoindToHom φ B A f) x) =
    x