English
There is a canonical bijection between Hom((sheafify α φ), F) and M0 ⟶ (restrictScalars α).obj F when F is a sheaf.
Русский
Для шарфификации существует биекция между Hom((sheafify α φ), F) и M0 ⟶ (restrictScalars α).obj F, если F — она.
LaTeX
$$$((sheafify α φ) ⟶ F) \cong (M_0 \to (\restrictScalars(α).obj F))$ underIsSheaf condition.$$
Lean4
/-- The bijection `((sheafify α φ).val ⟶ F) ≃ (M₀ ⟶ (restrictScalars α).obj F)` which
is part of the universal property of the sheafification of the presheaf of modules `M₀`,
when `F` is a presheaf of modules which is a sheaf. -/
noncomputable def sheafifyHomEquiv' {F : PresheafOfModules.{v} R.val} (hF : Presheaf.IsSheaf J F.presheaf) :
((sheafify α φ).val ⟶ F) ≃ (M₀ ⟶ (restrictScalars α).obj F) :=
(restrictHomEquivOfIsLocallySurjective α hF).trans
(homEquivOfIsLocallyBijective (f := toSheafify α φ) (N := (restrictScalars α).obj F) hF)
