Tensor $2$-Product for $\mathfrak{sl}_{2}$: Extensions to the Negative Half

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.