  We construct an explicit abelian model for the operation of tensor
$2$-product of $2$-representations of $\mathfrak{sl}_{2}^{+}$, 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.
