mem_ofArrows_iff — Mathlib

English

If α = β for functors F,G with NatTrans, then α.app X = β.app X for every X.

Русский

Если α = β для функторов F,G между ними NatTrans, то α.app X = β.app X для каждого X.

LaTeX

$$$\alpha_X = \beta_X$$$

Lean4

theorem mem_ofArrows_iff {W : C} (g : W ⟶ X) : ofArrows Y f g ↔ ∃ (i : I) (a : W ⟶ Y i), g = a ≫ f i :=
  by
  constructor
  · rintro ⟨T, a, b, ⟨i⟩, rfl⟩
    exact ⟨i, a, rfl⟩
  · rintro ⟨i, a, rfl⟩
    apply downward_closed _ (ofArrows_mk Y f i)

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