@Matthew – I don't understand this line:

> Based on [Lecture 60](https://forum.azimuthproject.org/discussion/2287/lecture-60-chapter-4-closed-monoidal-posets), to make \\(\mathrm{hom}\\) a \\(\mathcal{V}\\)-enriched functor we want \\(\mathcal{Y} = - \multimap - \\).

Surely we just set \\(\mathcal{Y}\\) to be \\(\mathcal{V}\\) (considered as a \\(\mathcal{V}\\)-enriched category)?

EDIT to add: oh wait, I see, you mean the function \\((y, y') \mapsto \mathcal{Y}(y, y')\\) should be \\((- \multimap -)\\)

EDIT to add: in which case I think this line is wrong:

> \[ (\mathcal{X}^{\text{op}} \times \mathcal{X})((a,b),(c,d)) = \mathcal{X}(b,a) \otimes \mathcal{X}(c,d) \]

It should read

\[ (\mathcal{X}^{\text{op}} \times \mathcal{X})((a,b),(c,d)) = \mathcal{X}^\text{op}(a, c) \otimes \mathcal{X}(b, d) = \mathcal{X}(c, a) \otimes \mathcal{X}(b, d) \]

... and then the proof drops out.

> Based on [Lecture 60](https://forum.azimuthproject.org/discussion/2287/lecture-60-chapter-4-closed-monoidal-posets), to make \\(\mathrm{hom}\\) a \\(\mathcal{V}\\)-enriched functor we want \\(\mathcal{Y} = - \multimap - \\).

Surely we just set \\(\mathcal{Y}\\) to be \\(\mathcal{V}\\) (considered as a \\(\mathcal{V}\\)-enriched category)?

EDIT to add: oh wait, I see, you mean the function \\((y, y') \mapsto \mathcal{Y}(y, y')\\) should be \\((- \multimap -)\\)

EDIT to add: in which case I think this line is wrong:

> \[ (\mathcal{X}^{\text{op}} \times \mathcal{X})((a,b),(c,d)) = \mathcal{X}(b,a) \otimes \mathcal{X}(c,d) \]

It should read

\[ (\mathcal{X}^{\text{op}} \times \mathcal{X})((a,b),(c,d)) = \mathcal{X}^\text{op}(a, c) \otimes \mathcal{X}(b, d) = \mathcal{X}(c, a) \otimes \mathcal{X}(b, d) \]

... and then the proof drops out.