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

Tactic:
aesop