I think we can show that inequality holds for any commutative quantale, and not just for \\(\mathbf{Bool}\\).
We start from the definition of enriched functor, since the mapping \\(\Phi\\) is a \\(\mathcal{V}\\)-functor between the \\(\mathcal{V}\\)-categories \\(\mathcal{X}^{\text{op}} \times \mathcal{Y}\\) and \\(\mathcal{V}\\):

\[ (\mathcal{X}^{\text{op}} \times \mathcal{Y})((x, y), (x', y')) \le \mathcal{V}(\Phi(x, y), \Phi(x', y')). \]

We use the definition of the product of \\(\mathcal{V}\\)-categories to get

\[ \mathcal{X}^{\text{op}}(x, x') \otimes \mathcal{Y}(y, y') \le \mathcal{V}(\Phi(x, y), \Phi(x', y')) \]

and the definition of the opposite of \\(\mathcal{V}\\)-category to get

\[ \mathcal{X}(x', x) \otimes \mathcal{Y}(y, y') \le \mathcal{V}(\Phi(x, y), \Phi(x', y')). \]

By remark 2.86 in the book, we know that we can enrich \\(\mathcal{V}\\) in itself using the closed property

\[ \mathcal{V}(v, w) = v \multimap w \]

which gives

\[ \mathcal{X}(x', x) \otimes \mathcal{Y}(y, y') \le \Phi(x, y) \multimap \Phi(x', y') \]

By applying the definition of the closed element (or hom-object)

\[ \mathcal{X}(x', x) \otimes \mathcal{Y}(y, y') \otimes \Phi(x, y) \le \Phi(x', y') \]

and using commutativity of the quantale \\(\mathcal{V}\\), we finally show that

\[ \mathcal{X}(x', x) \otimes \Phi(x, y) \otimes \mathcal{Y}(y, y') \le \Phi(x', y'). \]