@Matthew – re the second bit of your puzzle, I think we've already shown that. **Puzzle 197** tells us that \$$\mathrm{hom} \colon \mathcal{X}^{\text{op}} \times \mathcal{X} \to \mathcal{V}\$$ is a \$$\mathcal{V}\$$-functor, which tells us that it is a \$$\mathcal{V}\$$-profunctor \$$\mathcal{X} \nrightarrow \mathcal{X}\$$. If we set \$$\mathcal{X} = \mathcal{V}\$$, then we get \$$\multimap\$$ is a \$$\mathcal{V}\$$-profunctor \$$\mathcal{V} \nrightarrow \mathcal{V}\$$ as desired.