-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_5 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_6 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        d)
      (tensorRight v)
F G U : Functor.{v₁, v₂, u₁, u₂} C V
inst_9 : DayConvolutionUnit U
inst_10 : DayConvolution F U
inst_11 : DayConvolution U G
inst_12 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 U G _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F U _) G
inst_14 : DayConvolution F G
this :
  ((F ⊠ U) ⊠ G).IsLeftKanExtension (L :=
    Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C)) (F :=
    (F ⊠ Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) ⊠ G)
    (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G)
x : (C × Discrete PUnit.{1}) × C
hγ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) (G.obj x.2) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 F G _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))) x.2))
    (((𝟙 (F.obj x.1.1) ▷ tensorUnit.{v₂, u₂} V (𝒞 := inst_1) ≫ F.obj x.1.1 ◁ can) ▷ G.obj x.2 ≫
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ◁ 𝟙 (G.obj x.2)) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          ((@convolution _ _ _ _ inst_2 inst_3 F U _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
          (G.obj x.2))
        (NatTrans.app (unit F U) (x.1.1, tensorUnit.{v₁, u₁} C (𝒞 := inst)) ▷ G.obj x.2 ≫
          (@convolution _ _ _ _ inst_2 inst_3 F U _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ◁
            𝟙 (G.obj x.2))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
              x.2))
          (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
            (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)), x.2))
          (CategoryStruct.comp (Y :=
            (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 U G _) _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
                x.2))
            (NatTrans.app (Iso.hom (associator F U G))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
                x.2))
            (NatTrans.app (map (𝟙 F) (Iso.hom (DayConvolutionUnit.leftUnitor U G)))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
                x.2)))))
    (((𝟙 (F.obj x.1.1) ▷ tensorUnit.{v₂, u₂} V (𝒞 := inst_1) ≫ F.obj x.1.1 ◁ can) ▷ G.obj x.2 ≫
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ◁ 𝟙 (G.obj x.2)) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          ((@convolution _ _ _ _ inst_2 inst_3 F U _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
          (G.obj x.2))
        (NatTrans.app (unit F U) (x.1.1, tensorUnit.{v₁, u₁} C (𝒞 := inst)) ▷ G.obj x.2 ≫
          (@convolution _ _ _ _ inst_2 inst_3 F U _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ◁
            𝟙 (G.obj x.2))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
              x.2))
          (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
            (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)), x.2))
          (NatTrans.app (map (Iso.hom (DayConvolutionUnit.rightUnitor U F)) (𝟙 G))
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
              x.2))))
⊢ Eq (α :=
    ((F ⊠ Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) ⊠ G).obj x ⟶
      (Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C) ⋙
            Functor.prod (tensor C) (𝟭 C) ⋙ tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F G _).obj
        x)
    (NatTrans.app
      (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G ≫
        Functor.whiskerLeft
          (Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C))
          (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F U _) (unit F U) G ≫
            Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
              (CategoryStruct.comp (Y :=
                tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
                (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
                (Functor.whiskerLeft (tensor C)
                  (CategoryStruct.comp (Y :=
                    @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 U G _) _)
                    (Iso.hom (associator F U G)) (map (𝟙 F) (Iso.hom (DayConvolutionUnit.leftUnitor U G))))))))
      x)
    (NatTrans.app
      (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G ≫
        Functor.whiskerLeft
          (Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C))
          (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F U _) (unit F U) G ≫
            Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
              (CategoryStruct.comp (Y :=
                tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
                (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
                (Functor.whiskerLeft (tensor C) (map (Iso.hom (DayConvolutionUnit.rightUnitor U F)) (𝟙 G))))))
      x)

Tactic:
dsimp