  The quantum loop algebra $U_{v}(\mathcal{L}\mathfrak{g})$ was defined as a
generalization of the Drinfeld's new realization of the quantum affine algebra
to the loop algebra of any Kac-Moody algebra $\mathfrak{g}$. It has been shown
by Schiffmann that the Hall algebra of the category of coherent sheaves on a
weighted projective line is closely related to the quantum loop algebra
$U_{v}(\mathcal{L}\mathfrak{g})$, for some $\mathfrak{g}$ with a star-shaped
Dynkin diagram. In this paper we study Drinfeld's presentation of
$U_{v}(\mathcal{L}\mathfrak{g})$ in the double Hall algebra setting, based on
Schiffmann's work. We explicitly find out a collection of generators of the
double composition algebra $\mathbf{DC}(\Coh(\mathbb{X}))$ and verify that they
satisfy all the Drinfeld relations.
