The contravariant powerset functor $P : C^{\text{op}} \rightarrow C$ is the canonical example of duality.[1]
There are several intereresting duality principles which amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $P(A+B) \cong PA \times PB$ and \begin{gather*} P([f,g] : A+B \rightarrow D) = \\ \langle Pf,Pg\rangle : PD \rightarrow P(A + B). \end{gather*} Similarly for injections to the coproduct: \begin{gather*} P(i_1 : A \rightarrow A+B) = \\ \pi_1 : P(A+B) \rightarrow PA. \end{gather*} By expanding $P(A+B) \cong PA \times PB$ these are seen to be the normal product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar.
[1] Lawvere, Rosebrugh: "Sets for mathematics"
