-- 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
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : MonoidalCategoryStruct.{v₃, u₃} D
inst_6 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d : D
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)
fst : Discrete PUnit.{1}
x : C
a :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
    ((can ▷ Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x ≫
        Functor.obj.{v₁, v₂, u₁, u₂}
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ◁
          𝟙 (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)) ≫
      CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
        (NatTrans.app
          (DayConvolution.unit
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
          (tensorUnit.{v₁, u₁} C (𝒞 := inst), x))
        (NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (λ_ d)))
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x)))
    ((can ▷ Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x ≫
        Functor.obj.{v₁, v₂, u₁, u₂}
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ◁
          𝟙 (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)) ≫
      CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
        (NatTrans.app
          (DayConvolution.unit
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
          (tensorUnit.{v₁, u₁} C (𝒞 := inst), x))
        (NatTrans.app
          (Iso.hom
            (DayConvolutionUnit.leftUnitor
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)))
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x)))
⊢ Eq.{v₂ + 1} (α :=
    Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠
          Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (fst, x) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.prod (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙
          tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (fst, x))
    (NatTrans.app
      ((Functor.homEquivOfIsLeftKanExtension
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)) ⊠
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
            (ExternalProduct.extensionUnitLeft
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
              (φ (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4))))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
            (tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
          (a : (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d) →
            Functor.fromPUnit (𝟙_ V) ⊠ (ι C V D).obj d ⟶
              (Functor.fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj d)
        ((Functor.homEquivOfIsLeftKanExtension
              (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) _)
              (DayConvolution.unit
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
            (a : (ι C V D).obj (𝟙_ D) ⊛ (ι C V D).obj d ⟶ (ι C V D).obj d) →
              (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d)
          (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (λ_ d)))))
      (fst, x))
    (NatTrans.app
      ((Functor.homEquivOfIsLeftKanExtension
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)) ⊠
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
            (ExternalProduct.extensionUnitLeft
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
              (φ (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4))))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
            (tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
          (a : (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d) →
            Functor.fromPUnit (𝟙_ V) ⊠ (ι C V D).obj d ⟶
              (Functor.fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj d)
        ((Functor.homEquivOfIsLeftKanExtension
              (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) _)
              (DayConvolution.unit
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
            (a : (ι C V D).obj (𝟙_ D) ⊛ (ι C V D).obj d ⟶ (ι C V D).obj d) →
              (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d)
          (Iso.hom
            (DayConvolutionUnit.leftUnitor
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)))))
      (fst, x))

Tactic:
dsimp [corepresentableByLeft]

Premises used: CategoryTheory.MonoidalCategory.DayConvolutionUnit.corepresentableByLeft