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} \$