colimitCompWhiskeringLeftIsoCompColimit

English

The injection after whiskering left through G equals the whiskering-left of the colimit injection.

Русский

Инъекция после отбивки слева через G равна отбивке слева от самой инъекции колимита.

LaTeX

$$$\mathrm{colimit}.ι\, (F \!\circ (\mathrm{whiskeringLeft}\ D K \! C))\!_j \, \circ \, (\mathrm{colimitCompWhiskeringLeftIsoCompColimit}\, F \ G)^{\mathrm{hom}} = \mathrm{whiskerLeft}\ G (\mathrm{colimit}.ι F j)$$$

Lean4

/-- Taking a colimit after whiskering by `G` is the same as using `G` and then taking a colimit. -/
def colimitCompWhiskeringLeftIsoCompColimit (F : J ⥤ K ⥤ C) (G : D ⥤ K) [HasColimitsOfShape J C] :
    colimit (F ⋙ (whiskeringLeft _ _ _).obj G) ≅ G ⋙ colimit F :=
  NatIso.ofComponents
    (fun j =>
      colimitObjIsoColimitCompEvaluation (F ⋙ (whiskeringLeft _ _ _).obj G) j ≪≫
        HasColimit.isoOfNatIso (isoWhiskerLeft F (whiskeringLeftCompEvaluation G j)) ≪≫
          (colimitObjIsoColimitCompEvaluation F (G.obj j)).symm)

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