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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?

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    $\begingroup$ Not sure if this is what you are after, but you can make a category from $(P,L,I)$ by letting $P\cup L$ be the objects and the only nontrivial morphisms are dictated by members of $I$ (so $(p,l)\in I$ makes an arrow $p\to l$). Then its dual category is $(L,P,I^*)$. $\endgroup$ Feb 21, 2021 at 11:57

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