-- 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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_7 : DayConvolution F U
⊢ ∀ {Y Y' : Functor.{v₁, v₂, u₁, u₂} C V} (g : Y ⟶ Y')
    (f : @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _ ⟶ Y),
    Eq (α :=
      (obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
            obj (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V)
                (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) ⋙
              obj coyoneda (Opposite.op (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))))).obj
        Y')
      ((((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).homEquivOfIsLeftKanExtension
                  (DayConvolution.unit F U) Y').trans
              ((F ⊠ U).homEquivOfIsLeftKanExtension (extensionUnitRight U (φ U) F) (tensor C ⋙ Y')) :
            (a : F ⊛ U ⟶ Y') →
              ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                    (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj ((𝟭 C).prod (fromPUnit (𝟙_ C))) ⋙
                      coyoneda.obj (Opposite.op (F ⊠ fromPUnit (𝟙_ V)))).obj
                Y')
          (f ≫ g) :
        (obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
              obj (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V)
                  (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) ⋙
                obj coyoneda (Opposite.op (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))))).obj
          Y')
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
                    obj (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V)
                        (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) ⋙
                      obj coyoneda (Opposite.op (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))))).map
                g) :
            (a :
                (fun (X : Type (max u₁ v₂)) ↦ X)
                  (((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                        (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj ((𝟭 C).prod (fromPUnit (𝟙_ C))) ⋙
                          coyoneda.obj (Opposite.op (F ⊠ fromPUnit (𝟙_ V)))).obj
                    Y)) →
              (fun (X : Type (max u₁ v₂)) ↦ X)
                (((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                      (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj ((𝟭 C).prod (fromPUnit (𝟙_ C))) ⋙
                        coyoneda.obj (Opposite.op (F ⊠ fromPUnit (𝟙_ V)))).obj
                  Y'))
          ((((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).homEquivOfIsLeftKanExtension
                    (DayConvolution.unit F U) Y).trans
                ((F ⊠ U).homEquivOfIsLeftKanExtension (extensionUnitRight U (φ U) F) (tensor C ⋙ Y)) :
              (a : F ⊛ U ⟶ Y) →
                ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                      (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj ((𝟭 C).prod (fromPUnit (𝟙_ C))) ⋙
                        coyoneda.obj (Opposite.op (F ⊠ fromPUnit (𝟙_ V)))).obj
                  Y)
            f) :
        (fun (X : Type (max u₁ v₂)) ↦ X)
          ((obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
                obj (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V)
                    (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) ⋙
                  obj coyoneda (Opposite.op (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))))).obj
            Y'))

Tactic:
aesop