-- 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 : Finite J
f : J → C
X Y : C
val : Fintype J
intro :
  ∀ (j : J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) Y ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) 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) (f j) X) 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) (f j) X) Y)
                (rightDistributor f X ⊗ᵢ asIso (𝟙 Y) :
                  tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) Y ≅
                    tensorObj.{v_1, u_1} (𝒞 := inst)
                      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y)
                (rightDistributor (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y :
                  tensorObj.{v_1, u_1} (𝒞 := inst)
                      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y ≅
                    biproduct.{0, v_1, u_1} fun (j : J) ↦
                      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y)) :
            tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) Y ⟶
              biproduct.{0, v_1, u_1} fun (j : J) ↦
                tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y)
          (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) 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) (f j) X) Y) ⟶
              tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y) :
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) Y ⟶
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) 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) (f j) X) Y)
          (Iso.hom
            (α_ (biproduct.{0, v_1, u_1} f) X Y ≪≫
              Iso.trans (Y :=
                biproduct.{0, v_1, u_1} fun (j : J) ↦
                  tensorObj.{v_1, u_1} (𝒞 := inst) (f j) (tensorObj.{v_1, u_1} (𝒞 := inst) X Y))
                (rightDistributor f (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) :
                  tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) ≅
                    biproduct.{0, v_1, u_1} fun (j : J) ↦
                      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) (tensorObj.{v_1, u_1} (𝒞 := inst) X Y))
                (biproduct.mapIso fun (x : J) ↦ Iso.symm (α_ (f x) 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) (f j) (tensorObj.{v_1, u_1} (𝒞 := inst) X Y))
                    (biproduct.{0, v_1, u_1} fun (x : J) ↦
                      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f x) X) Y))))
          (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y)
            j) :
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) Y ⟶
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y)
⊢ Eq (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) Y ≅
      biproduct.{0, v_1, u_1} fun (j : J) ↦
        tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) 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) (f j) X) Y)
        (rightDistributor f X ⊗ᵢ asIso (𝟙 Y) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) Y ≅
            tensorObj.{v_1, u_1} (𝒞 := inst)
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y)
        (rightDistributor (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y :
          tensorObj.{v_1, u_1} (𝒞 := inst)
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y ≅
            biproduct.{0, v_1, u_1} fun (j : J) ↦
              tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) Y ≅
        biproduct.{0, v_1, u_1} fun (j : J) ↦
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) Y)
    (α_ (biproduct.{0, v_1, u_1} f) X Y ≪≫
        Iso.trans (Y :=
          biproduct.{0, v_1, u_1} fun (j : J) ↦
            tensorObj.{v_1, u_1} (𝒞 := inst) (f j) (tensorObj.{v_1, u_1} (𝒞 := inst) X Y))
          (rightDistributor f (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) :
            tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) ≅
              biproduct.{0, v_1, u_1} fun (j : J) ↦
                tensorObj.{v_1, u_1} (𝒞 := inst) (f j) (tensorObj.{v_1, u_1} (𝒞 := inst) X Y))
          (biproduct.mapIso fun (x : J) ↦ Iso.symm (α_ (f x) 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) (f j) (tensorObj.{v_1, u_1} (𝒞 := inst) X Y))
              (biproduct.{0, v_1, u_1} fun (x : J) ↦
                tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f x) X) Y)) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) Y ≅
        biproduct.{0, v_1, u_1} fun (x : J) ↦
          tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (f x) X) Y)

Tactic:
ext