Owen Biesel , that doesn't seem right actually.

You seem to be taking a scheme, which is a functor \\(F: \mathcal{C} \to \mathbf{Set}\\), and then applying it *backwards* along a functor \\(G\\) mapping between the categories \\(\mathcal{D} \to \mathcal{C}\\).

If intuition serves, \\(G\\) should mapping both tables into the single table. That is to say, \\(G\\) is an inclusion of sorts.

Therefor, the instance \\((F \circ G): \mathcal{D} \to \mathbf{Set}\\) should be one big table,

\\[

\begin{array}{c|c}

\text{People} & \mathrm{FriendOf} \\\\

\hline

Alice & Bob \\\\

Bob & Alice \\\\

\vdots & \vdots \\\\

Stan & Tyler \\\\

Tyler & Stan. \\\\

\vdots & \vdots \\\\

Adele & Sara\\\\

Bertram & Antonio \\\\

\vdots & \vdots \\\\

Siegmund & Teresa \\\\

\vdots & \vdots \\\\

Antonio & Bertram \\\\

Bruno & Bertram \\\\

\vdots & \vdots \\\\

Sara & Adele \\\\

Teresa & Siegmund. \\\\

\vdots & \vdots

\end{array}

\\]

Edit: Note that \\(Tyler\\) doesn't show up twice and that his Italian friend gets mapped to someone else as a friend.

You seem to be taking a scheme, which is a functor \\(F: \mathcal{C} \to \mathbf{Set}\\), and then applying it *backwards* along a functor \\(G\\) mapping between the categories \\(\mathcal{D} \to \mathcal{C}\\).

If intuition serves, \\(G\\) should mapping both tables into the single table. That is to say, \\(G\\) is an inclusion of sorts.

Therefor, the instance \\((F \circ G): \mathcal{D} \to \mathbf{Set}\\) should be one big table,

\\[

\begin{array}{c|c}

\text{People} & \mathrm{FriendOf} \\\\

\hline

Alice & Bob \\\\

Bob & Alice \\\\

\vdots & \vdots \\\\

Stan & Tyler \\\\

Tyler & Stan. \\\\

\vdots & \vdots \\\\

Adele & Sara\\\\

Bertram & Antonio \\\\

\vdots & \vdots \\\\

Siegmund & Teresa \\\\

\vdots & \vdots \\\\

Antonio & Bertram \\\\

Bruno & Bertram \\\\

\vdots & \vdots \\\\

Sara & Adele \\\\

Teresa & Siegmund. \\\\

\vdots & \vdots

\end{array}

\\]

Edit: Note that \\(Tyler\\) doesn't show up twice and that his Italian friend gets mapped to someone else as a friend.