I'm trying to prove the identity $\vert vec(AXB)\rangle = A\otimes B^T \vert vec(X)\rangle$, where $\vert vec(L)\rangle := \sum_{ij} L_{ij}\vert i\rangle\vert j\rangle$ for any $L:= \sum_{ij}L_{ij}\vert i\rangle\langle j\vert$.
The left hand side can be written as
$ \begin{align} (AXB)_{in} &= \sum_{jpqm}A_{ij}\vert i\rangle\langle j\vert X_{pq}\vert p\rangle\langle q\vert B_{mn}\vert m\rangle\langle n\vert \\ &=\sum_{jm}A_{ij}X_{jm}B_{mn}\vert i\rangle\langle n \vert \\ &\overset{\mathrm{vec}}{=} \sum_{jm}A_{ij}X_{jm}B_{mn}\vert i\rangle\vert n \rangle \end{align} $,
where I did the vectorization in the last line.
The right hand side is
$ \begin{align} (A\otimes B^T \vert vec(X)\rangle)_{in} &= \sum_{jpqm} A_{ij} \vert i\rangle\langle j \vert \otimes B_{nm}\vert n\rangle\langle m\vert\ X_{pq} \vert p\rangle\otimes\vert q\rangle \\ &= \sum_{jm} A_{ij}B_{nm}X_{jm} \vert i\rangle\vert n \rangle \end{align} $
These are not equal since $B_{mn} \neq B_{nm}$. Can anyone explain what I did wrong?