  We construct a Hopf algebra cocycle in the Yangian double $DY(SL_{2})$,
conjugating Drinfeld's coproduct to the usual one. To do that, we factorize the
twist between two ``opposite'' versions of Drinfeld's coproduct, introduced in
earlier work by V. Rubtsov and the first author, using the decomposition of the
algebra in its negative and non-negative modes subalgebras.
