  In a recent paper, the author defined an operation of tensor product for a
large class of $2$-representations of $\mathcal{U}^{+}$, the positive half of
the $2$-category associated to $\mathfrak{sl}_{2}$. In this paper, we prove
that the operation extends to give an operation of tensor product for
$2$-representations of the full $2$-category $\mathcal{U}$: when the inputs are
$2$-representations of the full $\mathcal{U}$, the $2$-product is also a
$2$-representation of the full $\mathcal{U}$. As in the previous paper, the
$2$-product is given for a simple $2$-representation $\mathcal{L}(1)$ and an
abelian $2$-representation $\mathcal{V}$ taken from the $2$-category of
algebras.
  This is the first construction of an operation of tensor product for higher
representations of a full Lie algebra in the abelian setting.
