-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Finite J
X : C
f : J → C
Y : C
intro :
  ∀ (val : Fintype J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ≅
        biproduct.{0, v_1, u_1} fun (j : J) ↦
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
      (Iso.trans (Y :=
          tensorObj.{v_1, u_1} (𝒞 := inst)
            (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
          (leftDistributor X f ⊗ᵢ asIso (𝟙 Y) :
            tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ≅
              tensorObj.{v_1, u_1} (𝒞 := inst)
                (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
          (rightDistributor (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y :
            tensorObj.{v_1, u_1} (𝒞 := inst)
                (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y ≅
              biproduct.{0, v_1, u_1} fun (j : J) ↦
                tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y) :
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ≅
          biproduct.{0, v_1, u_1} fun (j : J) ↦
            tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
      (α_ X (biproduct.{0, v_1, u_1} f) Y ≪≫
          Iso.trans (Y :=
            tensorObj.{v_1, u_1} (𝒞 := inst) X
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
            (asIso (𝟙 X) ⊗ᵢ rightDistributor f Y :
              tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) Y) ≅
                tensorObj.{v_1, u_1} (𝒞 := inst) X
                  (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
            (Iso.trans (Y :=
                biproduct.{0, v_1, u_1} fun (j : J) ↦
                  tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
                (leftDistributor X fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y :
                  tensorObj.{v_1, u_1} (𝒞 := inst) X
                      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) ≅
                    biproduct.{0, v_1, u_1} fun (j : J) ↦
                      tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
                (biproduct.mapIso fun (x : J) ↦ Iso.symm (α_ X (f x) Y) :
                  Iso.{v_1, u_1} (C := C)
                    (biproduct.{0, v_1, u_1} fun (j : J) ↦
                      tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
                    (biproduct.{0, v_1, u_1} fun (x : J) ↦
                      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f x)) Y)) :
              tensorObj.{v_1, u_1} (𝒞 := inst) X
                  (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) ≅
                biproduct.{0, v_1, u_1} fun (x : J) ↦
                  tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f x)) Y) :
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ≅
          biproduct.{0, v_1, u_1} fun (x : J) ↦
            tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f x)) Y)
intro₁ :
  ∀ (val : Fintype J) (j : J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
      (CategoryStruct.comp (Y :=
          biproduct.{0, v_1, u_1} fun (j : J) ↦
            tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
          (Iso.hom
              (Iso.trans (Y :=
                tensorObj.{v_1, u_1} (𝒞 := inst)
                  (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
                (leftDistributor X f ⊗ᵢ asIso (𝟙 Y) :
                  tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ≅
                    tensorObj.{v_1, u_1} (𝒞 := inst)
                      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
                (rightDistributor (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y :
                  tensorObj.{v_1, u_1} (𝒞 := inst)
                      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y ≅
                    biproduct.{0, v_1, u_1} fun (j : J) ↦
                      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)) :
            tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ⟶
              biproduct.{0, v_1, u_1} fun (j : J) ↦
                tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
          (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y) j :
            (biproduct.{0, v_1, u_1} fun (j : J) ↦
                tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y) ⟶
              tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y) :
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ⟶
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
      (CategoryStruct.comp (Y :=
          biproduct.{0, v_1, u_1} fun (j : J) ↦
            tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
          (Iso.hom
            (α_ X (biproduct.{0, v_1, u_1} f) Y ≪≫
              Iso.trans (Y :=
                tensorObj.{v_1, u_1} (𝒞 := inst) X
                  (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
                (asIso (𝟙 X) ⊗ᵢ rightDistributor f Y :
                  tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) Y) ≅
                    tensorObj.{v_1, u_1} (𝒞 := inst) X
                      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
                (Iso.trans (Y :=
                    biproduct.{0, v_1, u_1} fun (j : J) ↦
                      tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
                    (leftDistributor X fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y :
                      tensorObj.{v_1, u_1} (𝒞 := inst) X
                          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) ≅
                        biproduct.{0, v_1, u_1} fun (j : J) ↦
                          tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
                    (biproduct.mapIso fun (x : J) ↦ Iso.symm (α_ X (f x) Y) :
                      Iso.{v_1, u_1} (C := C)
                        (biproduct.{0, v_1, u_1} fun (j : J) ↦
                          tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
                        (biproduct.{0, v_1, u_1} fun (x : J) ↦
                          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f x)) Y)) :
                  tensorObj.{v_1, u_1} (𝒞 := inst) X
                      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) ≅
                    biproduct.{0, v_1, u_1} fun (x : J) ↦
                      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f x)) Y)))
          (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
            j) :
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ⟶
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
intro₂ :
  ∀ (val : Fintype J) (j : J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
      (∑ x : J,
        (X ◁ biproduct.π f x ⊗ₘ 𝟙 Y) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v_1, u_1} (𝒞 := inst)
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
            (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) x ⊗ₘ 𝟙 Y)
            (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ▷ Y))
      (∑ x : J,
        Iso.hom (α_ X (biproduct.{0, v_1, u_1} f) Y) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v_1, u_1} (𝒞 := inst) X
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
            (𝟙 X ⊗ₘ
              biproduct.π f x ▷ Y ≫
                biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) x)
            (X ◁ biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) j ≫ Iso.inv (α_ X (f j) Y)))
intro₃ :
  ∀ (val : Fintype J) (j : J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
      (∑ x : J,
        ((𝟙 X ⊗ₘ biproduct.π f x) ⊗ₘ 𝟙 Y) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v_1, u_1} (𝒞 := inst)
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
            (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) x ⊗ₘ 𝟙 Y)
            (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ⊗ₘ 𝟙 Y))
      (∑ x : J,
        Iso.hom (α_ X (biproduct.{0, v_1, u_1} f) Y) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v_1, u_1} (𝒞 := inst) X
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
            (𝟙 X ⊗ₘ
              (biproduct.π f x ⊗ₘ 𝟙 Y) ≫
                biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) x)
            ((𝟙 X ⊗ₘ biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) j) ≫ Iso.inv (α_ X (f j) Y)))
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ≅
      biproduct.{0, v_1, u_1} fun (j : J) ↦
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
    (Iso.trans (Y :=
        tensorObj.{v_1, u_1} (𝒞 := inst)
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
        (leftDistributor X f ⊗ᵢ asIso (𝟙 Y) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ≅
            tensorObj.{v_1, u_1} (𝒞 := inst)
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
        (rightDistributor (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y :
          tensorObj.{v_1, u_1} (𝒞 := inst)
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y ≅
            biproduct.{0, v_1, u_1} fun (j : J) ↦
              tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ≅
        biproduct.{0, v_1, u_1} fun (j : J) ↦
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
    (α_ X (biproduct.{0, v_1, u_1} f) Y ≪≫
        Iso.trans (Y :=
          tensorObj.{v_1, u_1} (𝒞 := inst) X
            (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
          (asIso (𝟙 X) ⊗ᵢ rightDistributor f Y :
            tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) Y) ≅
              tensorObj.{v_1, u_1} (𝒞 := inst) X
                (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
          (Iso.trans (Y :=
              biproduct.{0, v_1, u_1} fun (j : J) ↦
                tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
              (leftDistributor X fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y :
                tensorObj.{v_1, u_1} (𝒞 := inst) X
                    (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) ≅
                  biproduct.{0, v_1, u_1} fun (j : J) ↦
                    tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
              (biproduct.mapIso fun (x : J) ↦ Iso.symm (α_ X (f x) Y) :
                Iso.{v_1, u_1} (C := C)
                  (biproduct.{0, v_1, u_1} fun (j : J) ↦
                    tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y))
                  (biproduct.{0, v_1, u_1} fun (x : J) ↦
                    tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f x)) Y)) :
            tensorObj.{v_1, u_1} (𝒞 := inst) X
                (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) Y) ≅
              biproduct.{0, v_1, u_1} fun (x : J) ↦
                tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f x)) Y) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ≅
        biproduct.{0, v_1, u_1} fun (x : J) ↦
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f x)) Y)
have intro₄ : ∀ (val : Fintype J) (j : J),
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) Y ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) Y)
    (∑ x : J, if h : x = j then (𝟙 X ⊗ₘ biproduct.π f x) ≫ eqToHom sorry ⊗ₘ 𝟙 Y else (𝟙 X ⊗ₘ biproduct.π f x) ≫ 0 ⊗ₘ 𝟙 Y)
    (∑ x : J,
      if h : x = j then
        Iso.hom (α_ X (biproduct.{0, v_1, u_1} f) Y) ≫
          (𝟙 X ⊗ₘ (biproduct.π f x ⊗ₘ 𝟙 Y) ≫ eqToHom sorry) ≫ Iso.inv (α_ X (f j) Y)
      else
        Iso.hom (α_ X (biproduct.{0, v_1, u_1} f) Y) ≫ (𝟙 X ⊗ₘ (biproduct.π f x ⊗ₘ 𝟙 Y) ≫ 0) ≫ Iso.inv (α_ X (f j) Y))
Tactic: classical
  cases nonempty_fintype J
  ext
  simp only [Category.comp_id, Category.assoc, eqToHom_refl, Iso.symm_hom, Iso.trans_hom,
    asIso_hom, comp_zero, comp_dite, Preadditive.sum_comp, Preadditive.comp_sum, sum_tensor,
    tensor_sum, comp_tensor_id, tensorIso_hom, leftDistributor_hom, rightDistributor_hom,
    biproduct.mapIso_hom, biproduct.ι_map, biproduct.ι_π, Finset.sum_dite_irrel,
    Finset.sum_dite_eq', Finset.sum_const_zero, Finset.mem_univ, if_true]
  simp_rw [← tensorHom_id, ← id_tensorHom]
  simp only [← comp_tensor_id, ← id_tensor_comp_assoc, Category.assoc, biproduct.ι_π, comp_dite,
    dite_comp, tensor_dite, dite_tensor]
  simp