  We use analogues of Enright's and Arkhipov's functors to determine the quiver
and relations for a category of $\mathfrak{sl}_2 \ltimes L(4)$-modules which
are locally finite (and with finite multiplicities) over $\mathfrak{sl}_2$. We
also outline serious obstacles to extend our result to $\mathfrak{sl}_2 \ltimes
L(k)$, for $k>4$.
