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.
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