[Background: I asked a vague question the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question]
Summary of question: the inclusions are a particularly "good" class of morphisms in the category of sets. I've written down a bunch of properties they have, and reserve the right to write down more. Are there other classes of morphisms which share these properties?
Technical point. in ZFC if you define a function to be a collection of ordered pairs $(x,f(x))$ then two functions with different codomains can be equal as sets (and hence as functions). In this question, when I talk about a morphism $f:X\to Y$ in the category of sets, I mean the data of $f$ and $X$ and $Y$, as is normal in category theory. A function has a well-defined domain and codomain.
Set-up
I am looking for the following piece of data. For each set $X$ I want a set $P(X)$ of injections $f_i:A_i\to X$ in the category of sets, called the "good morphisms to $X$", with the following properties.
1) [representing injections]. For every injective map $g:Y\to X$ in the category of sets, there is a unique good $f:A\to X$ such that $g$ is isomorphic to $f$ in the sense that there's an isomorphism $Y\to A$ which makes the obvious triangle commute.
For conditions (2) and (3) we have injections $f:A\to B$ and $g:B\to C$ with composition $h:=g\circ f:A\to C$.
2) [closure] If $f:A\to B$ is good and $g:B\to C$ is good then $h := g\circ f:A\to C$ is good.
3) [g,h good implies f good] If $h:A\to C$ and $g:B\to C$ are both good, then $f$ is good too.
An example of a good class of maps is the set of all inclusions $i:A\to B$ where $A$ is a subset of $B$.
The question
The question (for which the answer is surely "yes of course") is: are there any other ways to choose a good class of maps with these properties?
I hesitate to put any more conditions, for example conditions about products of maps, because an object like $X\times Y$ is only defined up to unique isomorphism in the category of sets. This question is not at all "natural" in the category-theory sense, because if I replace my category by an equivalent category then I can't easily move my data from one to the other (as far as I can see). On the other hand, there are "canonical" (whatever that weasel word means!) constructions of products and limits in the category of sets, so I reserve the right to add more conditions about the behaviour of "nice" maps under limits if this question gets spiked too easily. In fact the more I think about it the more I wonder whether adding more criteria using these "canonical" constructions of limits (as an actual subset of a product, with the crucial observation being that subsets are being used) can actually turn this question into one with a positive answer, i.e. classifying the inclusions as those morphisms satisfying a bunch of properties, not all of them as "canonical" as one might like…
NB the word "canonical" does not have a definition in my mind, and mathematicians sometimes use it in a way where it can actually be replaced by a formal definition, but sometimes they use it to mean something which just looks like a good idea. I am trying to work out if inclusions are "canonical" monomorphisms, and this is a great example of a poor usage of the word in the sense that once you start digging you realise that you cannot supply a definition. I am attempting to supply a definition and still strongly suspect that I have failed.
Best Answer
The consensus seems to be that this is an answer to the question as stated (though I didn't originally realize that it was), so I'll go ahead and post it as one.
There are other ways to choose such a class of "good maps". For instance, you can transfer any class of good maps (such as the "standard" one consisting of inclusions) across an isomorphism of categories ${\rm Set} \cong {\rm Set}$ (though not necesarily across an equivalence of categories). To produce a nontrivial isomorphism ${\rm Set} \cong {\rm Set}$, choose two sets $X$ and $Y$ and an isomorphism $e:X≅Y$ that is not the identity. Define $F:Set→Set$ by $F(X)=Y$, $F(Y)=X$, and $F(A)=A$ for all other sets $A$, and define the action of $F$ on arrows by composing with the chosen isomorphism $e$ whenever needed. Then $F$ is an isomorphism ${\rm Set} \cong {\rm Set}$, under which a subset inclusion $i:X′↪X$ (for $X' \neq X,Y$) is sent to an injection $ei:X′→Y$ that is not (usually) an inclusion. More generally, you can construct an automorphism of ${\rm Set}$ that essentially arbitrarily permutes each bijection-equivalence-class of sets.
A way to make the question more interesting, as suggested by Peter Lumsdaine and Asaf Karagila, is to ask whether there is a class of good maps that is not related to the inclusions by such an automorphism of ${\rm Set}$, or similarly that is not equivalent as an M-category to the standard M-category of sets, functions, and inclusions. I don't know the answer to this version of the question.