prodComparison_natural — Mathlib

English

The morphism prodComparison is natural in both arguments; i.e., for f:g the corresponding diagram commutes with F.map and the tensor functor.

Русский

Морфизм prodComparison естествен по обеим аргументам; диаграмма коммутирует с F.map и тензорной функторной частью.

LaTeX

$$prodComparison_natural F A B f g$$

Lean4

/-- Naturality of the `prodComparison` morphism in both arguments. -/
@[reassoc]
theorem prodComparison_natural (f : A ⟶ A') (g : B ⟶ B') :
    F.map (f ⊗ₘ g) ≫ prodComparison F A' B' = prodComparison F A B ≫ (F.map f ⊗ₘ F.map g) := by
  apply hom_ext <;>
    simp only [Category.assoc, prodComparison_fst, tensorHom_fst, prodComparison_fst_assoc, prodComparison_snd,
      tensorHom_snd, prodComparison_snd_assoc, ← F.map_comp]

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