I was thinking... if functions can be described using a table of sets,

\\[
\begin{array}{c|c}
X & f \\\\
\hline
x_1 & f(x_1) \\\\
x_2 & f(x_2) \\\\
x_3 & f(x_3) \\\\
\vdots & \vdots
\end{array}
\\]

since functors map functions, would functors be described using a table of tables?

\\[
\begin{array}{c||c}
\mathcal{C} & F \\\\
\hline \hline
\begin{array}{c|c}
X & f \\\\
\hline
x_1 & f(x_1) \\\\
x_2 & f(x_2) \\\\
x_3 & f(x_3) \\\\
\vdots & \vdots
\end{array} & \begin{array}{c|c}
F(X) & F(f) \\\\
\hline
F(x_1) & F(f)(x_1) \\\\
F(x_2) & F(f)(x_2) \\\\
F(x_3) & F(f)(x_3)\\\\
\vdots & \vdots
\end{array} \\\\
\begin{array}{c|c}
Y & g \\\\
\hline
y_1 & g(y_1) \\\\
y_2 & g(y_2) \\\\
y_3 & g(y_3) \\\\
\vdots & \vdots
\end{array} & \begin{array}{c|c}
F(Y) & F(g) \\\\
\hline
F(y_1) & F(g)(y_1) \\\\
F(y_2) & F(g)(y_2) \\\\
F(y_3) & F(g)(y_3)\\\\
\vdots & \vdots
\end{array} \\\\
\vdots & \vdots
\end{array}
\\]