Oh, I think I see what I did wrong.

I made a functor from \$$\mathcal{C} \to \mathbf{Set}\$$ instead of \$$\mathcal{C}^{op} \times \mathcal{C} \to \mathbf{Set}\$$.

On a side note, since \$$\mathcal{C}^{op} \times \mathcal{C}\$$ is a category, we must also have a functor,
\$(\mathcal{C}^{op} \times \mathcal{C})^{op} \times \mathcal{C}^{op} \times \mathcal{C} \to \mathbf{Set}\\\\ =\mathcal{C} \times \mathcal{C}^{op} \times \mathcal{C}^{op} \times \mathcal{C} \to \mathbf{Set}. \$

We can keep doing this construction ad infinitum.

Also, on another side note, since \$$\mathrm{hom} : \mathcal{C}^{op} \times \mathcal{C} \to \mathbf{Set}\$$ is a functor to \$$\mathbf{Set}\$$, \$$\mathrm{hom}\$$ counts as a database instance, however it is one that comes automatic with every category.