Let $\mathcal{C}$ be a category, and let $X$ be an object of $\mathcal{C}$. A subobject of $X$ is by definition an equivalence class of pairs $(Y,i)$, where $Y$ is another object and $i\colon Y\to X$ is a monomorphism. Here $(Y_0,i_0)$ and $(Y_1,i_1)$ are considered equivalent if there is an isomorphism $f\colon Y_0\to Y_1$ with $i_1f=i_0$.
Now take $\mathcal{C}$ to be the category of compact Hausdorff spaces. The claim is that the subobjects of $X$ biject with the closed subsets.
The first point is to understand the monomorphisms in $\mathcal{C}$. By definition, a monomorphism is a continuous map $i\colon Y\to X$ of compact Hausdorff spaces such that $i_*\colon\mathcal{C}(T,Y)\to\mathcal{C}(T,X)$ is injective for all $T\in\mathcal{C}$. If $i$ itself is injective, then this clearly holds. Conversely, if $i$ is a monomorphism, we can take $T$ to be a one-point space, and we find that $i$ must be injective.
Next, it is a standard lemma that a subset of a compact Hausdorff space is closed iff it is compact in the subspace topology. If $i\colon Y\to X$ is a continuous map between compact Hausdorff spaces, then it certainly sends compact subsets to compact subsets, so it also sends closed subsets to closed subsets. Using this, we see that if $i$ is injective, then $i(Y)$ is closed, and if we give $i(Y)$ the subspace topology, then the map $i\colon Y\to i(Y)$ is a homeomorphism. Thus, in the class of subobjects, we have $[Y,i]=[i(Y),\text{inc}]$. This makes it clear that the subobjects of $X$ biject with the closed subsets.
As the question has been edited to ask about general categorical properties of $\mathcal{C}$, I will mention some more:
- Manes' Theorem: $\mathcal{C}$ is equivalent to the category of algebras for the Stone-Cech monad. Many properties stated below follow immediately from this, but they can also be proved more directly.
- $\mathcal{C}$ has all set-indexed limits and colimits.
- The monomorphisms are precisely the injective morphisms, and they are all equalisers.
- The epimorphisms are precisely the surjective morphisms, and they are all coequalisers.
- If $E\subseteq X\times X$ is an equivalence relation that is also closed as a subset of $X\times X$, then $X/E$ is a compact Hausdorff space, and is the coequaliser in $\mathcal{C}$ of the two evident maps $E\to X$.
- The injective objects are precisely the retracts of powers of the unit interval.
- An object $X$ is projective iff it is extremally disconnected, i.e. the closure of every open set is open.
- The dual category $\mathcal{C}^{\text{op}}$ is equivalent to a certain full subcategory of rings. The relevant rings are ordered, but we do not need to consider that as an extra ingredient, because it is determined by the ring structure: we have $a\leq b$ iff $b-a$ is a square. The relevant rings are also topologised, but we do not need to consider that as an extra ingredient, because it is determined by the order. Because of this, each of the relevant rings has a unique $\mathbb{R}$-algebra structure.
Many of these facts can be found in Johnstone's book "Stone Spaces".
