Categorical duality is the process of reversing all arrows. For example, the dual statement of $f: a \to b$ is $f: b \to a$; the dual statement of “$f$ is monic” is “$f$ is epi”, and the dual of “$t$ is terminal” is “$t$ is initial”.
The duality principle also applies to statements involving several categories and functors between them. Note that the duality here reverses the arrows in the category, not the functor between them.