Module: Mathlib.CategoryTheory.Monoidal.DayConvolution

Goal state before:
pp: C : Type u₁
inst✝¹⁴ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹³ : Category.{v₂, u₂} V
inst✝¹² : MonoidalCategory C
inst✝¹¹ : MonoidalCategory V
inst✝¹⁰ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝⁹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝⁸ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
inst✝⁷ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝⁶ :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow ((𝟭 C).prod (Functor.fromPUnit (𝟙_ C))) d) (tensorRight v)
F G U : C ⥤ V
inst✝⁵ : DayConvolutionUnit U
inst✝⁴ : DayConvolution F U
inst✝³ : DayConvolution U G
inst✝² : DayConvolution F (U ⊛ G)
inst✝¹ : DayConvolution (F ⊛ U) G
inst✝ : DayConvolution F G
⊢ extensionUnitLeft (F ⊛ U) (unit F U) G ≫
      ((tensor C).prod (𝟭 C)).whiskerLeft
        (unit (F ⊛ U) G ≫
          (tensor C).whiskerLeft ((associator F U G).hom ≫ map (𝟙 F) (DayConvolutionUnit.leftUnitor U G).hom)) =
    extensionUnitLeft (F ⊛ U) (unit F U) G ≫
      ((tensor C).prod (𝟭 C)).whiskerLeft
        (unit (F ⊛ U) G ≫ (tensor C).whiskerLeft (map (DayConvolutionUnit.rightUnitor U F).hom (𝟙 G)))

Tactic:
have : (F ⊠ U) ⊠ G |>.IsLeftKanExtension
      (α := extensionUnitLeft (F ⊠ U) (extensionUnitRight U (DayConvolutionUnit.φ U) F) G) :=
    isPointwiseLeftKanExtensionExtensionUnitLeft (F ⊠ U) _ G
      (isPointwiseLeftKanExtensionExtensionUnitRight U (DayConvolutionUnit.φ U) F <|
        DayConvolutionUnit.isPointwiseLeftKanExtensionCan (F := U)) |>.isLeftKanExtension

Goal state after:
C : Type u₁
inst✝¹⁴ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹³ : Category.{v₂, u₂} V
inst✝¹² : MonoidalCategory C
inst✝¹¹ : MonoidalCategory V
inst✝¹⁰ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝⁹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝⁸ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
inst✝⁷ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝⁶ :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow ((𝟭 C).prod (Functor.fromPUnit (𝟙_ C))) d) (tensorRight v)
F G U : C ⥤ V
inst✝⁵ : DayConvolutionUnit U
inst✝⁴ : DayConvolution F U
inst✝³ : DayConvolution U G
inst✝² : DayConvolution F (U ⊛ G)
inst✝¹ : DayConvolution (F ⊛ U) G
inst✝ : DayConvolution F G
this : ((F ⊠ U) ⊠ G).IsLeftKanExtension (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G)
⊢ extensionUnitLeft (F ⊛ U) (unit F U) G ≫
      ((tensor C).prod (𝟭 C)).whiskerLeft
        (unit (F ⊛ U) G ≫
          (tensor C).whiskerLeft ((associator F U G).hom ≫ map (𝟙 F) (DayConvolutionUnit.leftUnitor U G).hom)) =
    extensionUnitLeft (F ⊛ U) (unit F U) G ≫
      ((tensor C).prod (𝟭 C)).whiskerLeft
        (unit (F ⊛ U) G ≫ (tensor C).whiskerLeft (map (DayConvolutionUnit.rightUnitor U F).hom (𝟙 G)))