-- 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
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)
have intro : 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)))
Tactic: 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]