Given two matrices $A$ and $B$ of size $a \times n$ and $b \times m$ consider the following operation $A \dagger B$ whose result is an $a b^n \times n m$ matrix. $A \dagger B$ is a block matrix with $a$ blocks $(A \dagger B)_i$ of size $b^n \times n m$ each constructed as follows. For each vector $j \in \{1, \dots, b\}^n$ create a row $(A_{i,1} \cdot B_{j[1]}, A_{i,2} \cdot B_{j[2]}, \dots, A_{i,n} \cdot B_{j[n]})$ where $B_t$ is the $t$-th row of $B$.

Is there a name for this matrix operation?