-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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 : Fintype J
X : C
f : J → C
w :
  ∀ (j : J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
            tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶
              biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (Iso.inv (leftDistributor X f) :
            (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
              tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
          (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ≫ X ◁ biproduct.ι f j) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
w₁ :
  ∀ (j : J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
          (biproduct.desc fun (j : J) ↦ X ◁ biproduct.ι f j) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
          (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ≫ X ◁ biproduct.ι f j) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
⊢ Eq.{v_1 + 1} (α :=
    (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (Iso.inv (leftDistributor X f) :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ≫ X ◁ biproduct.ι f j :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))

Tactic:
classical
  ext
  dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]
  simp only [Preadditive.comp_sum, biproduct.ι_π_assoc, dite_comp, zero_comp,
    Finset.sum_dite_eq, Finset.mem_univ, ite_true, eqToHom_refl, Category.id_comp,
    biproduct.ι_desc]

Premises used: Finset.mem_univ, CategoryTheory.Limits.biproduct.ι_π_assoc, CategoryTheory.Functor.mapBicone, CategoryTheory.dite_comp, CategoryTheory.eqToHom_refl, ite_true, CategoryTheory.Limits.biproduct.ι_desc, CategoryTheory.Preadditive.comp_sum, CategoryTheory.Limits.zero_comp, Finset.sum_dite_eq, CategoryTheory.Functor.mapBiproduct, CategoryTheory.leftDistributor, CategoryTheory.Category.id_comp