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)
⊢ 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)) =
    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))

Tactic:
apply Functor.hom_ext_of_isLeftKanExtension (α := extensionUnitLeft ((F ⊛ G) ⊠ H)
      (extensionUnitLeft _ (unit F G) H) K)

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)
⊢ 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))) =
    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)))