What does "dual statement" mean exactly in category theory?

Some of the categories in a statement are "variables" (e.g. "$C$"), and you should take the opposite of those categories but not any categories which are "constants" (e.g. $\text{Set}$). The point is that any category which is a "variable" is being quantified over, so you can always replace that category with an opposite category, in the same way that you can substitute anything you want for a variable in an identity.

For example, if a functor $F : C \to D$ is a left adjoint, then it preserves colimits. The dual of this statement is obtained by replacing $C$ and $D$ with opposites, since they are both "variables," and you get that if $F : C^{op} \to D^{op}$ is a left adjoint, then it preserves colimits. But this is equivalent to a statement about $C$ and $D$ themselves, which is that if $F^{op} : C \to D$ is a right adjoint, then it preserves limits.