You may have a point.

So then \$$G\$$ is more like a mapping between the underlying graphs of the categories than a mapping between databases themselves.

Though, \$$G\$$ can be used to *make* a database, but only after we compose it with some \$$F: \mathcal{C} \to \mathbf{Set}\$$ to get the database instance, \$$(F \circ G): \mathcal{D} \to \mathbf{Set}\$$.