A tensor 2-product of 2-representations of ${\mathfrak{s}\mathfrak{l}}_2^{+}$

We construct an explicit abelian model for the operation of tensor $2$-product of $2$-representations of ${\mathfrak{s}\mathfrak{l}}_2^{+}$, specifically the product of a simple $2$-representation $\mathcal{L}(1)$ with a given abelian $2$-representation $\mathcal{V}$ taken from the $2$-category of algebras. We study the case $\mathcal{V}=\mathcal{L}(1)$ in detail, and we show that the $2$-product in this case recovers the expected structure. Our construction partially verifies a conjecture of Rouquier from 2008.