  We compare the reduced Drinfeld doubles of the composition subalgebras of the
category of representations of the Kronecker quiver $\overr{Q}$ and of the
category of coherent sheaves on ${\mathbb P}^1$. Using this approach, we show
that the Drinfeld--Beck isomorphism for the quantized enveloping algebra
$U_v(\hat{\mathfrak{sl}}_2)$ is a corollary of an equivalence between the
derived categories $D^b\bigl(\Rep(\overr{Q})\bigr)$ and $D^b\bigl(\Coh({\mathbb
P}^1)\bigr)$. This technique also allows to reprove several technical results
on the integral form of $U_v(\hat{\mathfrak{sl}}_2)$.
