How to formulate a theorem about bijections between several sets I have several sets $A_i$ and bijections between them. (As stated in my theorem) no composition of these bijections produces a permutation of $A_i$ not equal to identity. So every bijection is identified by the pair of sets between which it acts.

It would be to cumbersome to denote every bijection with a special letter (such as $\Phi$). I want to write it in some consise way.

For example I could denote the bijection from $A_i$ to $A_j$ as $\phi_{A_i,A_j}$ but this is not formally right as I would first to prove $A_i\ne A_j$. Then I would denote it $\phi_{i,j}$ but this way I would need explicitly number my sets, but I'd better to use English names or maybe letters to denote the sets not numbers.

The best solution I found insofar is to denote every set with some letter and denote my bijections as $\phi_{\alpha,\beta}$, where $\alpha$ and $\beta$ determine some sets. This solution is not ideal, because it would involve for each considered set two different notations to denote it: say $A_1$ and $\alpha$.

This is insofar the best solution I know. But maybe somebody may suggest me a better language to formulate my theorem?
Each bijection $\phi$ determines its domain and range. Why not phrase your theorem in terms of the indexed set of bijections $\phi_j$ where $j$ ranges over some arbitrary index set $I$? You can define $D_j$ and $R_j$ (say) to be the domain and range of $\phi_j$ respectively. The sets $D_j$ and $R_j$ are your $A_i$ but you don't have to worry about whether they are distinct.
The best idea I got is the following:

Consider only bijections of the set $A_0$ with $A_i$ for every $i\ne 0$. There are no need to explicitly say about bijections of every $A_i$ with $A_j$.

The rest bijections can be easily figured out by a reader of my writing.