pullbackConeOfLeftLift_snd — Mathlib

English

The inverse of the restriction construction equals the identity on the restricted presheaf.

Русский

Обратное ограничение совпадает с тождественным отображением на ограниченном предшехе.

LaTeX

$$ofRestrict_invApp X h U = 𝟙 _$$

Lean4

theorem pullbackConeOfLeftLift_snd : pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).snd = s.snd := by
  -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): `ext` did not pick up `NatTrans.ext`

  refine PresheafedSpace.Hom.ext _ _ ?_ <| NatTrans.ext <| funext fun x => ?_
  · change pullback.lift _ _ _ ≫ pullback.snd _ _ = _
    simp
  · change (_ ≫ _ ≫ _) ≫ _ = _
    simp_rw [Category.assoc]
    erw [s.snd.c.naturality_assoc]
    erw [← s.pt.presheaf.map_comp, ← s.pt.presheaf.map_comp]
    trans s.snd.c.app x ≫ s.pt.presheaf.map (𝟙 _)
    · congr 1
    · simp

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