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1) Justify each of the four steps \( (=, \le, \le, =) \) in Eq. (4.27).

2) In the case \( \mathcal{V} = \textbf{Bool} \), we can directly show each of the four steps in Eq. (4.27) is actually an equality. How?

**Equation 4.27**

Φ(p, q) I ⊗ Φ(p, q) ≤ P(p, p) ⊗ Φ(p, q) ≤ Ü P(p, p 1 ) ⊗ Φ(p 1 , q) (U P .Φ)(p, q).

## Comments

\(\Phi(p,q)=I\otimes\Phi(p,q)\) holds because \(\mathcal{V}\) is a symmetric monoidal preorder (in particular, this is property b) in the definition of a symmetric monoidal structure on a preorder). \(I\otimes\Phi(p,q)\leq\mathcal{P}(p,p)\otimes\Phi(p,q)\) holds since \(\mathcal{P}\) is a \(\mathcal{V}-\)category (in particular, property a) in the definition of a \(\mathcal{V}-\)category and property a) in the definition of a monoidal structure on a preorder). \(\mathcal{P}(p,p)\otimes\Phi(p,q)\leq\bigvee_{p_1\in\mathcal{P}}\mathcal{P}(p,p_1)\otimes\Phi(p_1,q)\) is true since \(p\in\mathcal{P}\), by the definition of join (joins always exist since \(\mathcal{V}\) is a quantale). \(\bigvee_{p_1\in\mathcal{P}}\mathcal{P}(p,p_1)\otimes\Phi(p_1,q)=(U_\mathcal{P}\circ\Phi)(p,q)\) by the definition of the unit profunctor and the definition of profunctor composition.

If \(\Phi(p,q)=\mathrm{true}\) then, since \(\mathrm{true}\) is the top element of

Bool, all of the other profunctors must also evaluate to \(\mathrm{true}\), since there is nothing larger for them to be. So all the inequalities are equalities. The only other possible value for \((U_\mathcal{P}\circ\Phi)(p,q)\) is \(\mathrm{false}\), which is the bottom element ofBool, so the other profunctors must also be \(\mathrm{false}\), as there is nothing smaller for them to be. So, again all the inequalities are equalities. Since these two possibilities cover all possible values of the profunctors in each part of the chain of equalities and inequalities, all of the relations in the chain are equalities when \(\mathcal{V}=\textbf{Bool}\).`1. \\(\Phi(p,q)=I\otimes\Phi(p,q)\\) holds because \\(\mathcal{V}\\) is a symmetric monoidal preorder (in particular, this is property b) in the definition of a symmetric monoidal structure on a preorder). \\(I\otimes\Phi(p,q)\leq\mathcal{P}(p,p)\otimes\Phi(p,q)\\) holds since \\(\mathcal{P}\\) is a \\(\mathcal{V}-\\)category (in particular, property a) in the definition of a \\(\mathcal{V}-\\)category and property a) in the definition of a monoidal structure on a preorder). \\(\mathcal{P}(p,p)\otimes\Phi(p,q)\leq\bigvee\_{p\_1\in\mathcal{P}}\mathcal{P}(p,p\_1)\otimes\Phi(p\_1,q)\\) is true since \\(p\in\mathcal{P}\\), by the definition of join (joins always exist since \\(\mathcal{V}\\) is a quantale). \\(\bigvee\_{p\_1\in\mathcal{P}}\mathcal{P}(p,p\_1)\otimes\Phi(p\_1,q)=(U\_\mathcal{P}\circ\Phi)(p,q)\\) by the definition of the unit profunctor and the definition of profunctor composition. 2. If \\(\Phi(p,q)=\mathrm{true}\\) then, since \\(\mathrm{true}\\) is the top element of **Bool**, all of the other profunctors must also evaluate to \\(\mathrm{true}\\), since there is nothing larger for them to be. So all the inequalities are equalities. The only other possible value for \\((U\_\mathcal{P}\circ\Phi)(p,q)\\) is \\(\mathrm{false}\\), which is the bottom element of **Bool**, so the other profunctors must also be \\(\mathrm{false}\\), as there is nothing smaller for them to be. So, again all the inequalities are equalities. Since these two possibilities cover all possible values of the profunctors in each part of the chain of equalities and inequalities, all of the relations in the chain are equalities when \\(\mathcal{V}=\textbf{Bool}\\).`

In the current version of Seven Sketches (in Compositionality), there is a third part:

3 Justify each of the three steps \((=,\leq,\leq)\newcommand{\cat}[1]{\mathcal{#1}}\) in \eqref{SecondHalfLeftUnitLaw}. \begin{equation}\label{SecondHalfLeftUnitLaw} \tag{4.29} \cat{P}(p,p_1)\otimes\Phi(p_1,q)=\cat{P}(p,p_1)\otimes\Phi(p_1,q)\otimes I\leq\cat{P}(p,p_1)\otimes\Phi(p_1,q)\otimes\cat{Q}(q,q)\leq\Phi(p,q).\end{equation}(Which the text says follows from Definitions 2.46 and 4.8.)

These follow from \(\mathcal{V}\) being a symmetric monoidal preorder (same property as in the above comment), \(\mathcal{Q}\) being a \(\mathcal{V}-\)category (same as above, which is Definition 2.46), and exercise 4.9 (which immediately follows definition 4.8 and is one of the equivalent ways of defining \(\mathcal{V}-\)profunctors).

`In the current version of Seven Sketches (in Compositionality), there is a third part: 3 Justify each of the three steps \\((=,\leq,\leq)\newcommand{\cat}[1]{\mathcal{#1}}\\) in \eqref{SecondHalfLeftUnitLaw}. \begin{equation}\label{SecondHalfLeftUnitLaw} \tag{4.29} \cat{P}(p,p_1)\otimes\Phi(p_1,q)=\cat{P}(p,p_1)\otimes\Phi(p_1,q)\otimes I\leq\cat{P}(p,p_1)\otimes\Phi(p_1,q)\otimes\cat{Q}(q,q)\leq\Phi(p,q).\end{equation}(Which the text says follows from Definitions 2.46 and 4.8.) These follow from \\(\mathcal{V}\\) being a symmetric monoidal preorder (same property as in the above comment), \\(\mathcal{Q}\\) being a \\(\mathcal{V}-\\)category (same as above, which is Definition 2.46), and exercise 4.9 (which immediately follows definition 4.8 and is one of the equivalent ways of defining \\(\mathcal{V}-\\)profunctors).`