# Pri is a reflective subcategory of Nach

(An application of this result is that [$Pri$ is order-regular](https://hackmd.io/@alexhkurz/ryPi3X5fL).) (CHECK)

Let

- $Pri$ be the category of Priestley spaces
- $Nach$ be the category of ordered compact Hausdorff spaces

with continuous and order-preserving maps.

**Theorem:** There is a dual adjunction
$$2^-\dashv 2^-: DL^{op}\to Nach$$

where $2$ is the two-chain, viewed as a distributed lattice in DL and equipped with the discrete toplogy in Nach.

**Proof:** This follows from the adjunction being induced by a [dualising object](https://hackmd.io/@alexhkurz/HyqDdTpNL).

**Remark:** Moreover, $2^-: DL^{op}\to Nach$ is fully faithful.