English
A detailed componentwise proof shows the limit property holds for the constructed cone.
Русский
Детальная компонентная часть доказательства показывает сохранение предельности для построенного конуса.
LaTeX
$$$\text{Let } c', f \text{ satisfy the required compatibilities, then } f \text{ is the lift for the limit cone.}$$$
Lean4
/-- The Fubini theorem for a functor `F : J ⥤ K ⥤ C`,
showing that the colimit of `uncurry.obj F` can be computed as
the colimit of the colimits of the functors `F.obj j`.
-/
noncomputable def colimitUncurryIsoColimitCompColim : colimit (uncurry.obj F) ≅ colimit (F ⋙ colim) :=
by
let c := colimit.cocone (uncurry.obj F)
let P : IsColimit c := colimit.isColimit _
let G := DiagramOfCocones.mkOfHasColimits F
let Q : ∀ j, IsColimit (G.obj j) := fun j => colimit.isColimit _
have Q' := coconeOfCoconeUncurryIsColimit Q P
have Q'' := colimit.isColimit (F ⋙ colim)
exact IsColimit.coconePointUniqueUpToIso Q' Q''
