Stone compactification vs initial/final structure as in Bourbaki Does the Stone-Cech compactification resemble final structure (such as quotient topology) as described in Bourbaki's Theory of Sets Ch.IV? I feel so since continuous $f:X \rightarrow K$ implies the existence of continuous $F:\beta X \rightarrow K$. Not sure I am on the right track though (probably not, but why?) I am not familiar with category theory so my feeling could be obviously wrong.
 A: This may not be what you're asking, but there is a sense in which the Stone-Cech compactification is "final" like the quotient topology... Unfortunately, the language is entirely muddled up because the initial structure category theoretically corresponds to having the most open sets you're allowed. Which means you're the biggest element in the lattice of possible topologies, and so you're called final.
Dually, of course, the topology a category theorist would call terminal or final is exactly the structure which is traditionally called initial. The language is entirely backwards. (If you'll continue indulging my rant -- this has permeated into the language of topologically concrete categories which is endlessly confusing).
To make the answer as reader-friendly as possible, I'll specify when I mean

*

*categorical-initial (there is a map to every other object)

*bourbaki-final (the same as above)

*categorical-final (there is a map from every other object)

*bourbaki-initial (same as above)

You need a mild technical assumption on $X$ (complete regularity), which I'll assume from here on out.

The reason you're seeing a similarity between $\beta X$ and the quotient topology is because they're both categorical-initial. Unfortunately, the quotient topology is called bourbaki-final, and $\beta X$ is typically called the "biggest" compactification... So at least they're both backwards.
To see why, we define a partial order on compactifications by setting
$$\alpha_2 \leq \alpha_1$$
whenever for some continuous $f$

That's not a typo: $\alpha_2 \leq \alpha_1$. The language is backwards.
If you like, this is the opposite poset structure that comes from the slice category under $X$. Since we're restricting attention to compactifications, and thus compact hausdorff spaces, we see $\alpha_1 \leq \alpha_2 \leq \alpha_1 \implies \alpha_1 X \cong \alpha_2 X$. So we really do get a poset.
Now one can show that $\beta$ is maximal in this poset. That is, it's categorical-initial in the category of compactifications under $X$ in a way that's entirely analogous to the quotient topology being categorical-initial in the category of "topologies identifying certain points" under $X$.

Rather interestingly, the one point compactification (when it exists), is smallest in this partial order. That is, the one point compactification is categorical-terminal, and is thus analogous to bourbaki-initial topologies in a similar sense.
You can find a proof of this here.
If you want an excellent blog post with proofs of the claims I glossed over, you should read this post from Dan Ma's topology blog. (You'll also notice I stole the above image from this site...). The whole series on the Stone-Cech compactification is good. In particular, though, you might find this post worthwhile too.

I hope this helps ^_^
A: Not really, it's functor from $\mathtt{Tych}$ to $\mathtt{CompHaus}$, so a map $f: X \to Y$ between Tychonoff spaces induces a unique $\beta f: \beta X \to \beta Y$ in a functiorial way (preserving compositions and identities). It is also has extension property as in AR (absolute retracts) theory.
And with a final or initial structure you start with a set and some maps, while with $\beta X$ you start with $X$ and construct both a new set and a new topology. There is no map.
Both things being "categorical" doesn't mean they're related.
