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?

> **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?