English
For the identity map id_X, the stalk pushforward germ aligns with the germ via the stalk functor: ℱ.stalkPushforward C (id X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom.
Русский
Для тождественного отображения id_X зародок стэла согласуется с зародком через стэковый функтор: ℱ.stalkPushforward C (id X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom.
LaTeX
$$$\mathcal{F}.stalkPushforward\ C\ (id_X)\ x = (stalkFunctor\ C\ x).map\ (Pushforward.id\ mathcal{F}).hom$$$
Lean4
@[simp]
theorem id (ℱ : X.Presheaf C) (x : X) : ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom :=
by
ext
simp only [stalkPushforward, germ, colim_map, ι_colimMap_assoc, whiskerRight_app]
erw [CategoryTheory.Functor.map_id]
simp [stalkFunctor]
