Module: Mathlib.CategoryTheory.Monoidal.DayConvolution

Goal state before:
pp: C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : MonoidalCategory V
F G : C ⥤ V
inst✝¹⁴ : DayConvolution F G
inst✝¹³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝¹¹ :
  ∀ (v : V) (d : C × C), Limits.PreservesColimitsOfShape (CostructuredArrow ((tensor C).prod (𝟭 C)) d) (tensorRight v)
H K : C ⥤ V
inst✝¹⁰ : DayConvolution G H
inst✝⁹ : DayConvolution (F ⊛ G) H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution H K
inst✝⁶ : DayConvolution G (H ⊛ K)
inst✝⁵ : DayConvolution (G ⊛ H) K
inst✝⁴ : DayConvolution ((F ⊛ G) ⊛ H) K
inst✝³ : DayConvolution (F ⊛ G) (H ⊛ K)
inst✝² : DayConvolution (F ⊛ G ⊛ H) K
inst✝¹ : DayConvolution F (G ⊛ H ⊛ K)
inst✝ : DayConvolution F ((G ⊛ H) ⊛ K)
⊢ map (associator F G H).hom (𝟙 K) ≫ (associator F (G ⊛ H) K).hom ≫ map (𝟙 F) (associator G H K).hom =
    (associator (F ⊛ G) H K).hom ≫ (associator F G (H ⊛ K)).hom

Tactic:
by
  -- We repeatedly apply the fact that the functors are left Kan extensions
  apply Functor.hom_ext_of_isLeftKanExtension (α := unit ((F ⊛ G) ⊛ H) K)
  apply Functor.hom_ext_of_isLeftKanExtension
    (α := extensionUnitLeft ((F ⊛ G) ⊛ H) (unit (F ⊛ G) H) K)
  have : (((F ⊛ G) ⊠ H) ⊠ K).IsLeftKanExtension
    (α := extensionUnitLeft ((F ⊛ G) ⊠ H)
      (extensionUnitLeft _ (unit F G) H) K) :=
    isPointwiseLeftKanExtensionExtensionUnitLeft _ _ _
      (isPointwiseLeftKanExtensionExtensionUnitLeft _ _ _
        (isPointwiseLeftKanExtensionUnit F G)) |>.isLeftKanExtension
  apply Functor.hom_ext_of_isLeftKanExtension (α := extensionUnitLeft ((F ⊛ G) ⊠ H)
      (extensionUnitLeft _ (unit F G) H) K)
  -- And then we compute...
  ext ⟨⟨⟨i, j⟩, k⟩, l⟩
  have aux :
      ((unit F G).app (i, j) ⊗ₘ (unit H K).app (k, l)) ≫
        (unit (F ⊛ G) (H ⊛ K)).app ((i ⊗ j), (k ⊗ l)) =
      (α_ (F.obj i) (G.obj j) (H.obj k ⊗ K.obj l)).hom ≫
        F.obj i ◁ G.obj j ◁ (unit H K).app (k, l) ≫ F.obj i ◁ (unit G (H ⊛ K)).app (j, (k ⊗ l)) ≫
        (unit F (G ⊛ H ⊛ K)).app (i, (j ⊗ k ⊗ l)) ≫ (F ⊛ G ⊛ H ⊛ K).map (α_ i j (k ⊗ l)).inv ≫
        (associator F G (H ⊛ K)).inv.app ((i ⊗ j) ⊗ k ⊗ l) := by
    conv_rhs => simp only [Functor.comp_obj, tensor_obj, NatTrans.naturality,
      associator_inv_unit_unit_assoc, externalProductBifunctor_obj_obj, Iso.map_hom_inv_id,
      Category.comp_id]
    simp only [tensor_whiskerLeft_symm, Category.assoc, Iso.hom_inv_id_assoc,
    ← tensorHom_def'_assoc]
  dsimp
  simp only [MonoidalCategory.whiskerLeft_id, Category.comp_id, unit_app_map_app_assoc,
    externalProductBifunctor_obj_obj, NatTrans.id_app, tensorHom_id, associator_hom_unit_unit_assoc,
    tensor_obj, NatTrans.naturality]
  conv_rhs =>
    simp only [whiskerRight_tensor_symm_assoc, Iso.inv_hom_id_assoc, ← tensorHom_def_assoc]
    rw [reassoc_of% aux]
  simp only [Iso.inv_hom_id_app_assoc, ← comp_whiskerRight_assoc, associator_hom_unit_unit F G H]
  simp only [Functor.comp_obj, tensor_obj, comp_whiskerRight, whisker_assoc, Category.assoc,
    whiskerRight_comp_unit_app_assoc (F ⊛ G ⊛ H) K l (α_ i j k).inv,
    NatTrans.naturality_assoc, NatTrans.naturality, associator_hom_unit_unit_assoc,
    externalProductBifunctor_obj_obj, unit_app_map_app_assoc, NatTrans.id_app, id_tensorHom,
    Iso.inv_hom_id_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc, associator_hom_unit_unit]
  simp [← Functor.map_comp, pentagon_inv, pentagon_assoc]

Goal state after: