“…Hence E 2 = 0 = W 2 , and W is the weak-injective envelope of A. Conversely, suppose that in the above diagram we start with the bottom exact sequence with (W , γ ) as the weak-injective envelope of A; the existence of such an exact sequence is guaranteed by Lee [9] or Göbel and Trlifaj [7]. Next we take the flat cover (E, β) of W ; in view of Lee [10], flat covers of weak-injectives are torsion-free injectives, so E is torsion-free injective. Let H = Ker β, and define α : F → A as the restriction of β.…”