@Keith – surely we already have an isomorphism of \$$\mathcal{V}\$$-categories \$$X \times Y \cong Y \times X\$$ given by \$$(x, y) \mapsto (y, x)\$$. We can check that this is in fact a \$$\mathcal{V}\$$-functor:

\$(Y \times X)((y, x), (y', x')) = Y(y, y') \otimes X(x, x') = X(x, x') \otimes Y(y, y') = (X \times Y)((x, y), (x', y')) \$

NB we need \$$\otimes\$$ to be symmetric in order to define the product \$$\mathcal{V}\$$-category \$$X \times Y\$$ in the first place.