@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.

\\[ (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.