[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. Just to be completely clear about this (because this might not be true in some ZFC ways of setting up the foundations), 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, so for example one cannot have two functions from the same domain being "technically equal" in the sense that they give the same outputs when presented with the same input, but they have different (overlapping) codomains. 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 morphisms $f_i:A_i\to X$ 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.