homEquiv_naturality_one — Mathlib

English

Symmetric iteration yields a dual form of the conjugate equivalence.

Русский

Симметричное повторение даёт двойственную форму сопряжённого эквивалента.

LaTeX

$$$\\text{iterated_mateEquiv_conjugateEquiv_symm}(α) = \\text{conjugateEquiv} (adj_1 \\circ adj_4) (adj_3 \\circ adj_2)^{-1} \\cdot α$$$

Lean4

@[reassoc]
theorem homEquiv_naturality_one (f₁ : X₁ ⟶ Y₁) (g : (F.obj Y₁).obj X₂ ⟶ X₃) :
    adj₂.homEquiv ((F.map f₁).app X₂ ≫ g) = adj₂.homEquiv g ≫ (G.map f₁.op).app X₃ :=
  by
  have := NatTrans.congr_app (adj₂.unit_whiskerRight_map f₁) X₂
  dsimp at this
  simp only [homEquiv_eq, Adjunction.homEquiv_unit, Functor.comp_obj, Functor.map_comp, Category.assoc,
    NatTrans.naturality, reassoc_of% this]

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