I was not understanding what was being 'enriched' and what was doing the 'enriching'. Based on comments I think I am not the only one. Here is my informal definition of \$$\mathcal{X} \$$ a \$$\mathcal{V}\$$-enriched category or \$$\mathcal{V}\$$-category. [Which may not turn out to be a category at all :P .]

You start with a container \$$\mathcal{X} \$$ which contains, in part, of a set of objects \$$\text{Ob}(\mathcal{X}) \$$. You want to say something about the structure of objects in this container. You decide that structure has something to do with the relationship between
pairs of objects \$$\mathcal{X}(x, y) | x, y \in Ob(\mathcal{X}) \$$ .

Stop for a moment and notice that we have not mentioned the enriching \$$\mathcal{V} \$$ yet.

We introduce the enriching \$$\mathcal{V} := (V, \le, I, \otimes ) \$$. Now we map each of the pairs \$$\mathcal{X}(x, y) \rightarrow V \$$ and in doing we say that \$$\mathcal{X} \$$ is enriched in \$$\mathcal{V} \$$.
[Why don't we say 'enriched by'?] Well almost enriched, the mapping is constrained by two properties.

a) We want the enrichment to reinforce the identity of each object. For the identity pair \$$\mathcal{X}(x, x) | x \in Ob( \mathcal{X} ) \$$ we have \$$I \le \mathcal{X}(x, x) \$$ where the \$$I\$$ and \$$\le \$$ come from \$$\mathcal{V} \$$.

b) We also want to have some notion of composition. When we have three objects \$$x, y, z \in \text{Ob}( \mathcal{X} ) \$$, with pairs \$$\mathcal{X}(x, y) \text{ and } \mathcal{X}(y, z) \$$. Using the order relation and monoidal operation \$$\le , \otimes \$$ from \$$\mathcal{V} \$$ we require that they compose in the following way \$$\mathcal{X}(x, y) \otimes \mathcal{X}(y, z) \le \mathcal{X}(x, z) \$$.

**Puzzle 87.** A preorder is isomorphic with **Bool**-category. What makes this odd it that the number of pairs in the **Bool**-category appears to be greater than the number of arrows in the corresponding preorder.
This discrepancy is explained by the fact that 'false' pairs are not expressed in the preorder.