Wiki. “Stone 对偶” [Stone对偶]

nLab

The two-element set carries a structure of Boolean algebra object in the category of compact Hausdorff spaces. So we have two algebraic structures on $\mathbb{F}_2$ whose operations commute with one another (here a compact Hausdorff space is an algebra for the ultrafilter monad, hence ‘algebraic’). This means two things: (1), that the hom-functor $CH(-, \mathbb{F}_2): CH^{op} \to Set$ is a Boolean algebra object, hence canonically lifts to a functor $CH^{op} \to Bool$. That’s one half of Stone duality. But equally well, $\mathbb{F}_2$ is a compact Hausdorff space object in the category of Boolean algebras. So (2), the hom-functor $Bool(-, \mathbb{F}_2): Bool^{op} \to Set$ is a compact Hausdorff space object, hence canonically lifts to a functor $Bool^{op} \to CH$. This is the other half of Stone duality. (All of this can be made set-theoretically rigorous.)