∀ {A : Type u_4} (K : Type u_2) (L : Type u_3) (L₂ : Type u_1) {B : Type u_5} {B₂ : Type u_6} [inst : CommRing A]
  [inst_1 : CommRing B] [inst_2 : CommRing B₂] [inst_3 : Algebra A B] [inst_4 : Algebra A B₂] [inst_5 : Field K]
  [inst_6 : Field L] [inst_7 : Field L₂] [inst_8 : Algebra A K] [inst_9 : IsFractionRing A K] [inst_10 : Algebra K L]
  [inst_11 : Algebra A L] [inst_12 : IsScalarTower A K L] [inst_13 : Algebra K L₂] [inst_14 : Algebra A L₂]
  [inst_15 : IsScalarTower A K L₂] [inst_16 : Algebra B L] [inst_17 : IsScalarTower A B L]
  [inst_18 : IsIntegralClosure B A L] [inst_19 : Algebra B₂ L₂] [inst_20 : IsScalarTower A B₂ L₂]
  [inst_21 : IsIntegralClosure B₂ A L₂] [inst_22 : Algebra.IsAlgebraic K L] [inst_23 : Algebra.IsAlgebraic K L₂]
  (σ : B ≃ₐ[A] B₂), (galLift K L L₂ ↑σ).comp (galLift K L₂ L ↑σ.symm) = AlgHom.id K L₂