Feas as as compact closed category

Let \(\mathcal{V}\) be a skeltal quantale. The category \(\mathbf{Prof}_\mathcal{V}\) can be given the structure of a compact closed category, with the monoidal product given by the product of \(\mathcal{V}\) categories.

Proof(1)

Monoidal product acts on objects:
- \(\mathcal{X} \times \mathcal{Y}((x,y),(x',y'))\) := \(\mathcal{X}(x,x') \otimes \mathcal{Y}(y,y')\)
Monoidal product acts on morphisms:
- \(\phi \times \psi((x_1,y_1),(x_2,y_2))\) := \(\phi(x_1,x_2)\otimes\psi(y_1,y_2)\)
Monoidal unit is the \(\mathcal{V}\) category \(1\)
Duals in \(\mathbf{Prof}_\mathcal{V}\) are just opposite categories
- For every \(\mathcal{V}\) category, \(\mathcal{X}\), its dual is \(\mathcal{X}^{op}\)
- The unit and counit look like identities
  - The unit is a \(\mathcal{V}\) profunctor \(1 \overset{\eta_\mathcal{X}}\nrightarrow \mathcal{X}^{op} \times \mathcal{X}\)
  - Alternatively \(1 \times \mathcal{X}^{op} \times \mathcal{X}\xrightarrow{\eta_\mathcal{X}}\mathcal{V}\)
  - Defined by \(\eta_\mathcal{X}(1,x,x'):=\mathcal{X}(x,x')\)
  - Likewise for the co-unit: \(\epsilon_\mathcal{X}(x,x',1):=\mathcal{X}(x,x')\)