Dear Phoenix87, I don't know where to write the answer to your latest question, so I put it here, though I'm afraid this will be interpreted as another answer to the original question. Anyway, two books where profinite monoids and semigroups are developed are: "Finite semigroups and universal algebra" by J. Almeida (in this book the term "profinite" is not used, yet such topics are treated) and "The q-theory of finite semigroups" by J. Rhodes and B. Steinberg.

Maybe I should make a further remark to your example which perhaps supports your understanding.

As I mentioned already, $S_0$ is the free proACom object on a single generator (lets call it $x_1$); more generally, $S_n$ is the free proACom object on a finite space consisting of $2^n$ elements, lets call this $X_n$ (the set consisting of $x_1,x_2,\dots, x_{2^n}$). Now the homomorphism $\phi_n:S_{n+1}\to S_n$ is completely determined by the values it takes on the free generators $X_{n+1}$ of $S_{n+1}$, these values are all in $X_n$ (free generators of $S_n$), according to your definition this mapping is just

$$x_1,x_2\mapsto x_1, x_3,x_4\mapsto x_2, x_5,x_6\mapsto x_3 \dots.$$

So you have an inverse system of finite discrete spaces

$$\dots \to X_{n+1}\to X_n\to X_{n-1}\to \dots\to X_0.$$

From the universal property of the projective limit it follows that the free object on the projective limit of the $X_n$s is just the projective limit of the free objects on the $X_n$s (you may interchange forming the projective limit and the free object). But the projective limit of the $X_n$s is just the Cantor set.