$\mathbf{Question:}$ Let $V$ be a finite-dimensional vector space over an algebraically closed field $F$. Fix $A,B \in \mathscr{L}(V)$. Consider the linear operator $T_{A,B} \in \mathscr{L}(\mathscr{L}(V))$ such that $T_{A,B}(X) = AX + XB$. Express $T_{A,B} $in terms of tensor products using the identification of $\mathscr{L}(V)$ with $V^*\otimes V$.
$\mathbf{Answer:} A \otimes I + I \otimes B^*$
Please explain. Every time I pick up this problem again I feel like I'm just blindly manipulating expressions. Thank you!