Let me say a bit more about this puzzle:

> **Puzzle 122.** Let \\(\mathcal{C}\\) be the free category on this graph:

>


> Let's use this as a database schema. Draw an example of a database built using this schema. Mathematically it's a functor \\(F:\mathcal{C} \to \textrm{Set}\\). But what does this mean in concrete terms? You can draw it as a table, like you would see in a spreadsheet.

Keith gave this answer:

\\[
\begin{array}{c|c}
\text{People} & \mathrm{FriendOf} \\\\
\hline
Alice & Bob \\\\
Bob & Alice \\\\
\vdots & \vdots \\\\
Stan & Tyler \\\\
Tyler & Stan \\\\
\vdots & \vdots
\end{array}
\\]

This table has a symmetry that's not required of every functor \\(F : \mathcal{C} \to \mathbf{Set}\\) Another perfectly fine answer would have been this:

\\[
\begin{array}{c|c}
\text{People} & \mathrm{FriendOf} \\\\
\hline
Alice & Bob \\\\
Bob & Bob \\\\
\vdots & \vdots \\\\
Stan & Bob \\\\
Tyler & Alice \\\\
\vdots & \vdots
\end{array}
\\]


**Puzzle.** Can we change the category \\(\mathcal{C}\\) to a category \\(\mathcal{C}^\prime\\) so that functors \\(F : \mathcal{C}^\prime \to \mathbf{Set}\\) are just databases of the sort Keith drew, with the kind of symmetry that his table has?