the category of T0 Alexandroff spaces is equivalent to the category of posets

Let $𝒜𝒯$ be the category of all ${\mathrm{T}}_0$, Alexandroff spaces and continuous maps between them. Furthermore let $𝒫𝒪𝒮ℰ𝒯$ be the category of all posets and order preserving maps.

Theorem. The categories $𝒜𝒯$ and $𝒫𝒪𝒮ℰ𝒯$ are equivalent.

Proof. Consider two functors:

$$T:𝒜𝒯\to 𝒫𝒪𝒮ℰ𝒯;$$

$$S:𝒫𝒪𝒮ℰ𝒯\to 𝒜𝒯,$$

such that $T(X,\tau )=(X,\le )$, where $\le$ is an induced partial order on an Alexandroff space and $T(f)=f$ for continuous map. Analogously, let $S(X,\le )=(X,\tau )$, where $\tau$ is an induced Alexandroff topology on a poset and $S(f)=f$ for order preserving maps. One can easily show that $T$ and $S$ are well defined. Furthermore, it is easy to verify that equalities $T\circ S=1_{𝒫𝒪𝒮ℰ𝒯}$ and $S\circ T=1_{𝒜𝒯}$ hold, which completes the proof. $\mathrm{\square }$

Remark. Of course every finite topological space is Alexandroff, thus we have very nice ,,interpretation” of finite ${\mathrm{T}}_0$ spaces - finite posets (since functors $T$ and $S$ do not change set-theoretic properties of underlying sets such as finitness).