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
⊢ (extensionUnitLeft ((F ⊛ G) ⊠ H) (extensionUnitLeft (F ⊛ G) (unit F G) H) K ≫
          (((tensor C).prod (𝟭 C)).prod (𝟭 C)).whiskerLeft
            (extensionUnitLeft ((F ⊛ G) ⊛ H) (unit (F ⊛ G) H) K ≫
              ((tensor C).prod (𝟭 C)).whiskerLeft
                (unit ((F ⊛ G) ⊛ H) K ≫
                  (tensor C).whiskerLeft
                    (map (associator F G H).hom (𝟙 K) ≫
                      (associator F (G ⊛ H) K).hom ≫ map (𝟙 F) (associator G H K).hom)))).app
      (((i, j), k), l) =
    (extensionUnitLeft ((F ⊛ G) ⊠ H) (extensionUnitLeft (F ⊛ G) (unit F G) H) K ≫
          (((tensor C).prod (𝟭 C)).prod (𝟭 C)).whiskerLeft
            (extensionUnitLeft ((F ⊛ G) ⊛ H) (unit (F ⊛ G) H) K ≫
              ((tensor C).prod (𝟭 C)).whiskerLeft
                (unit ((F ⊛ G) ⊛ H) K ≫
                  (tensor C).whiskerLeft ((associator (F ⊛ G) H K).hom ≫ (associator F G (H ⊛ K)).hom)))).app
      (((i, j), k), l)

Tactic:
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]

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)
⊢ (extensionUnitLeft ((F ⊛ G) ⊠ H) (extensionUnitLeft (F ⊛ G) (unit F G) H) K ≫
          (((tensor C).prod (𝟭 C)).prod (𝟭 C)).whiskerLeft
            (extensionUnitLeft ((F ⊛ G) ⊛ H) (unit (F ⊛ G) H) K ≫
              ((tensor C).prod (𝟭 C)).whiskerLeft
                (unit ((F ⊛ G) ⊛ H) K ≫
                  (tensor C).whiskerLeft
                    (map (associator F G H).hom (𝟙 K) ≫
                      (associator F (G ⊛ H) K).hom ≫ map (𝟙 F) (associator G H K).hom)))).app
      (((i, j), k), l) =
    (extensionUnitLeft ((F ⊛ G) ⊠ H) (extensionUnitLeft (F ⊛ G) (unit F G) H) K ≫
          (((tensor C).prod (𝟭 C)).prod (𝟭 C)).whiskerLeft
            (extensionUnitLeft ((F ⊛ G) ⊛ H) (unit (F ⊛ G) H) K ≫
              ((tensor C).prod (𝟭 C)).whiskerLeft
                (unit ((F ⊛ G) ⊛ H) K ≫
                  (tensor C).whiskerLeft ((associator (F ⊛ G) H K).hom ≫ (associator F G (H ⊛ K)).hom)))).app
      (((i, j), k), l)