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 $P(A+B) \cong PA \times PB$ and using the coproduct laws, these are seen to be the 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. This kind of application of the contravariant powerset functor can be done for all finite colimits.
For exponentials, the situation is much different. Trying to find operation $B \setminus A$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $P(B \setminus A) \cong PA \Rightarrow PB$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows seems obvious solution, but (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $F \dashv A + -$ (which might also be called subtraction if it exists).
[1] Lawvere, Rosebrugh: "Sets for mathematics"
[2] Crolard: Subtractive logic