A nice example of a functor naturally ismorphic to Stone-functor. I want to explain the natural transformation with an example involving the Stone functor, but, I can't think of any non-trivial one. Does any one have one? 
 A: (For those who, like me, did not know what the Stone functor was, this is the contravariant functor from topological spaces to Boolean algebras that assigns, to each topological space, the Boolean algebra of its clopen sets with union and intersection operations.)
There is a contravariant functor from topological spaces to commutative unital rings given by $X \mapsto C(X)$ that sends a topological space to its ring of continuous real-valued (or complex-valued, if you prefer) functions.  
There is a covariant functor from commutative unital rings to Boolean algebras, which sends a ring $R$ to the set of its idempotents, using the following operations on two idempotents $e, f \in R$:


*

*$e \vee f = e + f - ef$

*$e \wedge f = ef$

*The "bottom" and "top" elements are $0$ and $1$, respectively.


Now an exercise for you: prove that the Stone functor is naturally isomorphic to the composite $\mathbf{Top}^{op} \to \mathbf{cRing} \to \mathbf{Boolean}$ of the two functors defined above.  A healthy hint for you: the isomorphism between these two Boolean algebras should send a clopen subset $K \subseteq X$ with the characteristic function $\chi_K$ (whose value is $1$ on $K$ and $0$ on $X \setminus K$).
