English
The whiskering on the right inverts the colimit injection under the colimitComp iso: whiskerRight (colimit.ι G j) F ≫ (colimitCompWhiskeringRightIsoColimitComp F G).inv = colimit.ι (G ⋙ (whiskeringRight ...).obj F) j.
Русский
Правое взвешивание производит композицию так, чтобы whiskerRight(колимит.ι G j) F, умноженное на inv, давало колимит-ι.
LaTeX
$$$\\mathrm{whiskerRight}(\\operatorname{colimit.\\,ι} G j) F \\; \\circ (\\operatorname{colimitCompWhiskeringRightIsoColimitComp} F G)\\mathrm{inv} = \\operatorname{colimit.\\,ι}\\bigl(G \\!\\cdot\\! (\\mathrm{whiskeringRight}\\;\\_\\;\\_\\_).\\mathrm{obj}F\\bigr) j$$$
Lean4
@[reassoc (attr := simp)]
theorem whiskerRight_ι_colimitCompWhiskeringRightIsoColimitComp_inv {C : Type*} [Category C] {D : Type*} [Category D]
{E : Type*} [Category E] {J : Type*} [Category J] [HasColimitsOfShape J D] (F : D ⥤ E)
[PreservesColimitsOfShape J F] (G : J ⥤ C ⥤ D) (j : J) :
whiskerRight (colimit.ι G j) F ≫ (colimitCompWhiskeringRightIsoColimitComp F G).inv =
colimit.ι (G ⋙ (whiskeringRight _ _ _).obj F) j :=
by simp [Iso.comp_inv_eq]
