Matrix 2-vector spaces exhibit horizontal and vertical composition

Let $F:{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^2\to {\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^2$ have matrix $\left(\begin{array}{cc}\mathbb{C}& 0\\ \mathbb{C}& \mathbb{C}\end{array}\right)$ and let $G$ have matrix $\left(\begin{array}{cc}\mathbb{C}& \mathbb{C}\\ 0& \mathbb{C}\end{array}\right)$. With rows indexed by outputs and columns by inputs, $(GF{)}_{ki}={⨁}_j{G}_{kj}\otimes F_{ji}$, so $GF=\left(\begin{array}{cc}\mathbb{C}\oplus \mathbb{C}& \mathbb{C}\\ \mathbb{C}& \mathbb{C}\end{array}\right)\mathrm{.}$ Each summand is the contribution of one intermediate simple label.

Let $\eta :F\Rightarrow F^{\mathrm{\prime }}$ scale $F_{21}$ by $2$, let ${\eta }^{\mathrm{\prime }}:F^{\mathrm{\prime }}\Rightarrow F^{\mathrm{\prime }\mathrm{\prime }}$ scale the same entry by $7$, let $\theta :G\Rightarrow G^{\mathrm{\prime }}$ scale $G_{12}$ by $3$, and let ${\theta }^{\mathrm{\prime }}:G^{\mathrm{\prime }}\Rightarrow G^{\mathrm{\prime }\mathrm{\prime }}$ scale the same entry by $5$; all other entries and components are identities. On the second intermediate-label summand of the $(1,1)$ entry of $GF$, horizontal composition first gives scalar $3\cdot 2=6$, and the next horizontal 2-cell gives $5\cdot 7=35$, so vertical composition gives $35\cdot 6=210$. Vertically composing first gives scalar $5\cdot 3=15$ in $G_{12}$ and $7\cdot 2=14$ in $F_{21}$, whose horizontal tensor is $15\cdot 14=210$. The first intermediate-label summand is unchanged in both paths. Thus the two interchange composites agree entrywise, and hence agree as natural transformations.