∀ (p : ℕ) [inst : Fact (Nat.Prime p)] {R : Type u_1} [inst_1 : CommRing R] (I : Ideal R) [inst_2 : CharP (R ⧸ I) p]
  [inst_3 : IsAdicComplete I R],
  (Perfection.mapMonoidHom p ↑(Ideal.Quotient.mk I)).comp
      ((Perfection.liftMonoidHom p (Perfection (R ⧸ I) p) R) (Perfection.teichmuller p I)) =
    MonoidHom.id (Perfection (R ⧸ I) p)