In projective geometry, the principle of duality states that any theorem that holds for an incidence structure $(P, L, I)$, where $P$ are the points, $L$ are the lines and $I \subseteq P \times L$ is the incidence relation, also holds for the dual incidence structure $(L, P, I^*)$, where $I^*$ is the converse relation of $I$.
In category theory, the principle of duality states that a categorical theorem has a dual, which holds for the dual category obtained by reversing all the arrows.
How can the first one be specified in such a way that it derives from the second one?