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
⊢ ((𝟙 (F.obj x✝.1.1) ▷ 𝟙_ V ≫ F.obj x✝.1.1 ◁ can) ▷ G.obj x✝.2 ≫ (F.obj x✝.1.1 ⊗ U.obj (𝟙_ C)) ◁ 𝟙 (G.obj x✝.2)) ≫
      ((unit F U).app (x✝.1.1, 𝟙_ C) ▷ G.obj x✝.2 ≫ (F ⊛ U).obj (x✝.1.1 ⊗ 𝟙_ C) ◁ 𝟙 (G.obj x✝.2)) ≫
        (unit (F ⊛ U) G).app (x✝.1.1 ⊗ 𝟙_ C, x✝.2) ≫
          (associator F U G).hom.app ((x✝.1.1 ⊗ 𝟙_ C) ⊗ x✝.2) ≫
            (map (𝟙 F) (DayConvolutionUnit.leftUnitor U G).hom).app ((x✝.1.1 ⊗ 𝟙_ C) ⊗ x✝.2) =
    ((𝟙 (F.obj x✝.1.1) ▷ 𝟙_ V ≫ F.obj x✝.1.1 ◁ can) ▷ G.obj x✝.2 ≫ (F.obj x✝.1.1 ⊗ U.obj (𝟙_ C)) ◁ 𝟙 (G.obj x✝.2)) ≫
      ((unit F U).app (x✝.1.1, 𝟙_ C) ▷ G.obj x✝.2 ≫ (F ⊛ U).obj (x✝.1.1 ⊗ 𝟙_ C) ◁ 𝟙 (G.obj x✝.2)) ≫
        (unit (F ⊛ U) G).app (x✝.1.1 ⊗ 𝟙_ C, x✝.2) ≫
          (map (DayConvolutionUnit.rightUnitor U F).hom (𝟙 G)).app ((x✝.1.1 ⊗ 𝟙_ C) ⊗ x✝.2)

Tactic:
simp only [MonoidalCategory.whiskerRight_id, Category.id_comp, Iso.hom_inv_id, whisker_assoc,
    MonoidalCategory.whiskerLeft_id, Category.comp_id,
    DayConvolution.associator_hom_unit_unit_assoc, externalProductBifunctor_obj_obj, tensor_obj,
    NatTrans.naturality, unit_app_map_app_assoc, NatTrans.id_app, id_tensorHom,
    Category.assoc, Iso.inv_hom_id_assoc, unit_app_map_app, Functor.comp_obj,
    tensorHom_id, Iso.cancel_iso_hom_left]

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 ◁ 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)