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
this : ((F ⊠ U) ⊠ G).IsLeftKanExtension (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G)
x✝ : (C × Discrete PUnit.{1}) × C
⊢ F.obj x✝.1.1 ◁ can ▷ G.obj x✝.2 ≫
      F.obj x✝.1.1 ◁ (unit U G).app (𝟙_ C, x✝.2) ≫
        F.obj x✝.1.1 ◁ (DayConvolutionUnit.leftUnitor U G).hom.app (𝟙_ C ⊗ x✝.2) ≫
          (unit F G).app (x✝.1.1, 𝟙_ C ⊗ x✝.2) ≫ (F ⊛ G).map (α_ x✝.1.1 (𝟙_ C) x✝.2).inv =
    F.obj x✝.1.1 ◁ can ▷ G.obj x✝.2 ≫
      (α_ (F.obj x✝.1.1) (U.obj (𝟙_ C)) (G.obj x✝.2)).inv ≫
        (unit F U).app (x✝.1.1, 𝟙_ C) ▷ G.obj x✝.2 ≫
          (DayConvolutionUnit.rightUnitor U F).hom.app (x✝.1.1 ⊗ 𝟙_ C) ▷ G.obj x✝.2 ≫
            (unit F G).app (x✝.1.1 ⊗ 𝟙_ C, x✝.2)

Tactic:
simp only [← MonoidalCategory.whiskerLeft_comp_assoc, leftUnitor_hom_unit_app,
    associator_inv_naturality_middle_assoc, ← comp_whiskerRight_assoc, rightUnitor_hom_unit_app]

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)
x✝ : (C × Discrete PUnit.{1}) × C
⊢ F.obj x✝.1.1 ◁ ((λ_ (G.obj x✝.2)).hom ≫ G.map (λ_ x✝.2).inv) ≫
      (unit F G).app (x✝.1.1, 𝟙_ C ⊗ x✝.2) ≫ (F ⊛ G).map (α_ x✝.1.1 (𝟙_ C) x✝.2).inv =
    (α_ (F.obj x✝.1.1) (𝟙_ V) (G.obj x✝.2)).inv ≫
      ((ρ_ (F.obj x✝.1.1)).hom ≫ F.map (ρ_ x✝.1.1).inv) ▷ G.obj x✝.2 ≫ (unit F G).app (x✝.1.1 ⊗ 𝟙_ C, x✝.2)