Yours is the totient valence function, and it is a result of Erdős dating back to 1958 that if for some $m$ there exists at least one $n$ such that $|\phi^{-1}(n)|=m$ then $|\phi^{-1}(n)|=m$ for infinitely many $n$; see P. Erdős, Some Remarks on Euler's $\phi$-Function, Acta Math. (1958), Vol. 4, pp. 10-19.

Sorry, but this question is not appropriate to MO. What about $A = \mathbb N \setminus S$ with $S$ being any finite subset of $\mathbb N$?

@Ben: I may be missing something, but it seems to me that, while succeeding in the goal of generalizing the notion of "being isomorphic as objects", your idea fails to do the same with the notion of isomorphism, in that it deals only with arrows of type $f: X \to X$ (such that $f^2 = f$). Right?

@Todd: You have nothing to be sorry about. Indeed, thank you for your comments.

@Todd. Why not? What is wrong with semicats/semigroupoids? From a philosophical point of view, I've always believed - were it nothing but my silly faith - that minimalism, which is the name that I use to refer to the absence of conceptual redundancies, is the right way to go to shed light on sometimes obscure aspects of our most shining theories (and biases), in and out of the mathematical reality. In any case, you asked for a specific example. So, here it is: What should it be an isomorphism in the setting of semicats, as we cannot count on local identities? I've got my own idea, but...

Thank you, Benjamin, for your comments and the reference. I've just checked Tilson's paper: Appendix B indeed deals with semigroupoids but it is three pages long and, as you guessed, not really focused on what I am looking for.

Sure, but nonetheless I can't find anything close to my expectations, and this is why I resolved to ask here for advices. E.g., I'd like to know if semicategories have been previously considered by anyone with regard to the definition of a semantics for (finitary) first-order logics. But my concerns are even more primitive, let's say, and this is why I'm looking for a systematic development of the theory from the very basics.

For what it is worth, I encountered the first definition in a few papers. Also, I heard of it in a recent seminar, here in Paris, and I was a little disappointed by the answer of the speaker when I raised the question substantially phrased in the OP. On the other hand, I couldn't find a reference in Howie's or Clifford and Preston's books, so I thought to ask here. And it was a good idea, as is usual with MO.

Thank you, Benjamin. Let me just add a link to the arXiv preprint of your paper: arxiv.org/pdf/math/0611896v1.pdf. I put it on the top of the pile of must-read things.

I agree, my comment should be referred to both propositions, sorry for the lack of clarity. As for the second one, Clifford doesn't, however, seem to address torsion-freeness at all, unless I'm missing something.

@Dorian. For the record, your version of Bonnesen's inequality can be strenghtened by replacing the $\pi$ in the right-hand side with its square. Also, you should really define your $R_{\rm in}$ and $R_{\rm out}$ by a sup and an inf, respectively.