re **Puzzle 87**, @John mentioned that the "process of deducing" the answer is important... so I thought it might be instructive to simply replace \\(\mathcal{V}\\) with \\(\mathbf{Bool}\\) in his definition to see where it gets us:

> A **\\(\mathbf{Bool}\\)-enriched category** \\(\mathcal{X}\\) consists of two parts, satisfying two properties. First:

>

> 1. one specifies a set \\(\mathrm{Ob}(\mathcal{X})\\), elements of which are called **objects**;

>

> 2. for every two objects \\(x,y\\), one specifies an element \\(\mathcal{X}(x,y)\\) of \\(\mathbf{Bool}\\).

>

> Then:

>

> a) for every object \\(x\in\text{Ob}(\mathcal{X})\\) we require that

>

> \[ {\tt{true}} \leq \mathcal{X}(x, x) \]

>

> b) for every three objects \\(x,y,z\in\mathrm{Ob}(\mathcal{X})\\), we require that

>

> \[ \mathcal{X}(x,y)\wedge \mathcal{X}(y,z)\leq\mathcal{X}(x,z) \]

It's pretty clear that (a) is equivalent to \\[\mathcal{X}(x, x) = \tt{true}\\] and (b) is equivalent to \\[\mathcal{X}(x,y) = {\\tt{true}}

{\textrm{ and }} \mathcal{X}(y,z) = {\\tt{true}} {\textrm{ implies }} \mathcal{X}(x,z) = {\\tt{true}}\\] ie, the reflexive and transitive laws.

> A **\\(\mathbf{Bool}\\)-enriched category** \\(\mathcal{X}\\) consists of two parts, satisfying two properties. First:

>

> 1. one specifies a set \\(\mathrm{Ob}(\mathcal{X})\\), elements of which are called **objects**;

>

> 2. for every two objects \\(x,y\\), one specifies an element \\(\mathcal{X}(x,y)\\) of \\(\mathbf{Bool}\\).

>

> Then:

>

> a) for every object \\(x\in\text{Ob}(\mathcal{X})\\) we require that

>

> \[ {\tt{true}} \leq \mathcal{X}(x, x) \]

>

> b) for every three objects \\(x,y,z\in\mathrm{Ob}(\mathcal{X})\\), we require that

>

> \[ \mathcal{X}(x,y)\wedge \mathcal{X}(y,z)\leq\mathcal{X}(x,z) \]

It's pretty clear that (a) is equivalent to \\[\mathcal{X}(x, x) = \tt{true}\\] and (b) is equivalent to \\[\mathcal{X}(x,y) = {\\tt{true}}

{\textrm{ and }} \mathcal{X}(y,z) = {\\tt{true}} {\textrm{ implies }} \mathcal{X}(x,z) = {\\tt{true}}\\] ie, the reflexive and transitive laws.