English
For F : A ⥤ B and η : F ⟶ G with G,H : B ⥤ C, the whiskerRight left-right construction satisfies a naturality relation with mapWhiskerRight and mapCompLeft.
Русский
Для F : A ⥤ B и η : F ⟶ G с G,H : B ⥤ C, конструкция whiskerRight включает естественную естественную связь с mapWhiskerRight и mapCompLeft.
LaTeX
$$$\operatorname{mapWhiskerRight}(\whiskerLeft F η)(\mathbf{1}_D) = (\operatorname{mapCompLeft}\ D F G)^{\mathrm{hom}} \;\circ \; \mathrm{whiskerLeft}(\operatorname{mapPair}(F, \mathbf{1}_D))\; (\operatorname{mapWhiskerRight}\ η _) \; \circ (\operatorname{mapCompLeft}\ D F H)^{\mathrm{inv}}$$$
Lean4
@[reassoc]
theorem mapWhiskerRight_whiskerLeft (F : A ⥤ B) {G H : B ⥤ C} (η : G ⟶ H) :
mapWhiskerRight (whiskerLeft F η) (𝟭 D) =
(mapCompLeft D F G).hom ≫ whiskerLeft (mapPair F (𝟭 D)) (mapWhiskerRight η _) ≫ (mapCompLeft D F H).inv :=
by apply natTrans_ext <;> ext <;> simp [mapCompLeft]
