universal_property — Mathlib

English

If φ is an isomorphism, and p lifts f to φ, then φ is strongly cartesian.

Русский

Если φ изоморфизм и подъём p над f exists, то φ сильно-картезианский.

LaTeX

$$$[IsHomLift p f φ.hom] [IsIso φ] \\Rightarrow IsStronglyCartesian p f φ.hom$$$

Lean4

/-- The universal property of a strongly co-Cartesian morphism.

This lemma is more flexible with respect to non-definitional equalities than the field
`universal_property'` of `IsStronglyCocartesian`. -/
theorem universal_property {S' : 𝒮} {b' : 𝒳} (g : S ⟶ S') (f' : R ⟶ S') (hf' : f' = f ≫ g) (φ' : a ⟶ b')
    [IsHomLift p f' φ'] : ∃! χ : b ⟶ b', IsHomLift p g χ ∧ φ ≫ χ = φ' :=
  by
  subst_hom_lift p f' φ'; clear a b R S
  have : p.IsHomLift (f ≫ g) φ' := (hf' ▸ inferInstance)
  apply IsStronglyCocartesian.universal_property' f

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