conjugateIsoEquiv — Mathlib

English

Symmetric iteration preserves isomorphisms via conjugation.

Русский

Симметричное применение сохраняет изоморфизмы через сопряжения.

LaTeX

$$$\\text{iterated_mateEquiv_conjugateEquiv_symm}(α) = \\text{...}$$$

Lean4

/-- Thus conjugation defines an equivalence between natural isomorphisms. -/
@[simps]
def conjugateIsoEquiv : (L₂ ≅ L₁) ≃ (R₁ ≅ R₂)
    where
  toFun
    α :=
    { hom := conjugateEquiv adj₁ adj₂ α.hom
      inv := conjugateEquiv adj₂ adj₁ α.inv }
  invFun
    β :=
    { hom := (conjugateEquiv adj₁ adj₂).symm β.hom
      inv := (conjugateEquiv adj₂ adj₁).symm β.inv }
  left_inv := by cat_disch
  right_inv := by cat_disch

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