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)
this :
  (((F ⊛ G) ⊠ H) ⊠ K).IsLeftKanExtension (extensionUnitLeft ((F ⊛ G) ⊠ H) (extensionUnitLeft (F ⊛ G) (unit F G) H) K)
l k i j : C
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)
⊢ (((unit F G).app (i, j) ▷ H.obj k ≫ (F ⊛ G).obj (i ⊗ j) ◁ 𝟙 (H.obj k)) ▷ K.obj l ≫
        ((F ⊛ G).obj (i ⊗ j) ⊗ H.obj k) ◁ 𝟙 (K.obj l)) ≫
      ((unit (F ⊛ G) H).app (i ⊗ j, k) ▷ K.obj l ≫ ((F ⊛ G) ⊛ H).obj ((i ⊗ j) ⊗ k) ◁ 𝟙 (K.obj l)) ≫
        (unit ((F ⊛ G) ⊛ H) K).app ((i ⊗ j) ⊗ k, l) ≫
          (map (associator F G H).hom (𝟙 K)).app (((i ⊗ j) ⊗ k) ⊗ l) ≫
            (associator F (G ⊛ H) K).hom.app (((i ⊗ j) ⊗ k) ⊗ l) ≫
              (map (𝟙 F) (associator G H K).hom).app (((i ⊗ j) ⊗ k) ⊗ l) =
    (((unit F G).app (i, j) ▷ H.obj k ≫ (F ⊛ G).obj (i ⊗ j) ◁ 𝟙 (H.obj k)) ▷ K.obj l ≫
        ((F ⊛ G).obj (i ⊗ j) ⊗ H.obj k) ◁ 𝟙 (K.obj l)) ≫
      ((unit (F ⊛ G) H).app (i ⊗ j, k) ▷ K.obj l ≫ ((F ⊛ G) ⊛ H).obj ((i ⊗ j) ⊗ k) ◁ 𝟙 (K.obj l)) ≫
        (unit ((F ⊛ G) ⊛ H) K).app ((i ⊗ j) ⊗ k, l) ≫
          (associator (F ⊛ G) H K).hom.app (((i ⊗ j) ⊗ k) ⊗ l) ≫ (associator F G (H ⊛ K)).hom.app (((i ⊗ j) ⊗ k) ⊗ l)

Tactic:
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]

Goal state after:
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)
this :
  (((F ⊛ G) ⊠ H) ⊠ K).IsLeftKanExtension (extensionUnitLeft ((F ⊛ G) ⊠ H) (extensionUnitLeft (F ⊛ G) (unit F G) H) K)
l k i j : C
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)
⊢ (unit F G).app (i, j) ▷ H.obj k ▷ K.obj l ≫
      (unit (F ⊛ G) H).app (i ⊗ j, k) ▷ K.obj l ≫
        (associator F G H).hom.app ((i ⊗ j) ⊗ k) ▷ K.obj l ≫
          (unit (F ⊛ G ⊛ H) K).app ((i ⊗ j) ⊗ k, l) ≫
            (associator F (G ⊛ H) K).hom.app (((i ⊗ j) ⊗ k) ⊗ l) ≫
              (map (𝟙 F) (associator G H K).hom).app (((i ⊗ j) ⊗ k) ⊗ l) =
    (unit F G).app (i, j) ▷ H.obj k ▷ K.obj l ≫
      (α_ ((F ⊛ G).obj (i ⊗ j)) (H.obj k) (K.obj l)).hom ≫
        (F ⊛ G).obj (i ⊗ j) ◁ (unit H K).app (k, l) ≫
          (unit (F ⊛ G) (H ⊛ K)).app (i ⊗ j, k ⊗ l) ≫
            (associator F G (H ⊛ K)).hom.app ((i ⊗ j) ⊗ k ⊗ l) ≫ (F ⊛ G ⊛ H ⊛ K).map (α_ (i ⊗ j) k l).inv