mateEquiv_counit_symm — Mathlib

English

Mate equivalence commutes with vertical composition of squares.

Русский

Эквивалентность мейт сохраняет вертикальное композиционное квадрат.

LaTeX

$$$$(\\text{mateEquiv adj1 adj3})\\big(\\alpha \\; \\vcomp\\; \\beta\\big) = (\\text{mateEquiv adj1 adj2 }\\alpha) \\; \\vcomp \\; (\\text{mateEquiv adj2 adj3 }\\beta).$$$$

Lean4

/-- A component of a transposed version of the inverse mates correspondence. -/
theorem mateEquiv_counit_symm (α : TwoSquare R₁ H G R₂) (d : D) :
    L₂.map (α.app _) ≫ adj₂.counit.app _ = ((mateEquiv adj₁ adj₂).symm α).app _ ≫ H.map (adj₁.counit.app d) :=
  by
  conv_lhs => rw [← (mateEquiv adj₁ adj₂).right_inv α]
  exact
    (mateEquiv_counit adj₁ adj₂ ((mateEquiv adj₁ adj₂).symm α) d)
      /- A component of a transposed version of the mates correspondence. -/

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