-- Module: Mathlib.CategoryTheory.Distributive.Monoidal
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : MonoidalCategory.{v, u_1} C
inst_2 : HasBinaryCoproducts.{v, u_1} C
inst_3 : IsMonoidalLeftDistrib.{v, u_1} C
X Y Z : C
rw :
  Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
    (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
        (X ◁ coprod.inl : tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
        (CategoryStruct.comp (Y :=
            coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
            (Iso.inv (∂L X Y Z) :
              tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z) ⟶
                coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
            (hom (∂L X Y Z) :
              coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z) ⟶
                tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z)) :
          tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z) ⟶
            tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z)) :
      tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
    (CategoryStruct.comp (Y :=
        coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
        (coprod.inl :
          tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶
            coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
        (hom (∂L X Y Z) :
          coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z) ⟶
            tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z)) :
      tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
rw₁ :
  X ◁ coprod.inl ≫ 𝟙 (tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z)) =
    CategoryStruct.comp (Y := coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
      (coprod.inl :
        tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶
          coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
      (hom (∂L X Y Z) :
        coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z) ⟶
          tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
rw₂ :
  Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
    (X ◁ coprod.inl : tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
    (CategoryStruct.comp (Y :=
        coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
        (coprod.inl :
          tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶
            coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
        (hom (∂L X Y Z) :
          coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z) ⟶
            tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z)) :
      tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
rw₃ rw₄ :
  Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
    (X ◁ coprod.inl) (X ◁ coprod.inl)
⊢ Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
    (CategoryStruct.comp (Y :=
        coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
        (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
            (X ◁ coprod.inl :
              tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
            (Iso.inv (∂L X Y Z) :
              tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z) ⟶
                coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z)) :
          tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶
            coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
        (hom (∂L X Y Z) :
          coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z) ⟶
            tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z)) :
      tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))
    (CategoryStruct.comp (Y :=
        coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
        (coprod.inl :
          tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶
            coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z))
        (hom (∂L X Y Z) :
          coprod.{v, u_1} (tensorObj.{v, u_1} (𝒞 := inst) X Y) (tensorObj.{v, u_1} (𝒞 := inst) X Z) ⟶
            tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z)) :
      tensorObj.{v, u_1} (𝒞 := inst) X Y ⟶ tensorObj.{v, u_1} (𝒞 := inst) X (coprod.{v, u_1} Y Z))

Tactic:
rw [assoc, Iso.inv_hom_id, comp_id, coprod_inl_leftDistrib_hom]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.inv_hom_id, CategoryTheory.coprod_inl_leftDistrib_hom, CategoryTheory.Category.comp_id