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

Tactic:
ext