mateEquiv_eq_iff — Mathlib

English

Characterizes when two 2-cells correspond via the mates bijection given adjunctions adj₁ and adj₂.

Русский

Характеризуется равенство двух 2-кампазатиков через биекция mates при adjunctions adj₁ и adj₂.

LaTeX

$$$\\mathrm{mateEquiv\_eq\_iff} (\\alpha) (\\beta) \\;:\\; (\\mathrm{mateEquiv}\\ adj\\_1 adj\\_2 \\alpha = \\beta) \\iff (\\text{relation between adjunctions})$$$

Lean4

theorem mateEquiv_eq_iff (α : g ≫ l₂ ⟶ l₁ ≫ h) (β : r₁ ≫ g ⟶ h ≫ r₂) :
    mateEquiv adj₁ adj₂ α = β ↔ adj₁.homEquiv₁.symm β = adj₂.homEquiv₂ α ≫ (α_ _ _ _).hom :=
  by
  conv_lhs => rw [eq_comm, ← adj₁.homEquiv₁.symm.injective.eq_iff']
  rw [mateEquiv_apply, Equiv.symm_apply_apply]

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