Your post reminded me of mathoverflow.net/questions/5957/what-is-a-metric-space

@KCd Given the title, that question is far more interesting than I expected!

If you have "structure and properties [you] would like to have as the objects", then the morphisms should be "functions that respect the structure and properties." You may later discover you can ensure that with fewer conditions (groups really have three operations on them, not one, but asking functions to respect one suffices to ensure the other two), but that is essentially what you want.

In your final example, the "ambuguity" comes from whether you want to preserve the distance (isometries), or just the norm/continuity (bounded).

@ArturoMagidin I understand that the morphisms are intended to preserve the structure, but my question relates more to whether there should always be a clear choice when you have that goal in mind. For example, I don't think many people would question the wisdom of defining group homomorphisms the way they are defined. It seems like the only viable choice for preserving the group structure

@OliverHouse: That's not a question about categories, then, but about how you ensure that "preserve the structure" is achieved. And I would actually take issue with your assertion about groups: I think that the way group morphisms are usually defined (just preserve multiplication) is misleading and unwise. Group morphisms should be defined to preserve all the structure: map identity to identity, map products to products, and map inverses to inverses. It is then a theorem that all you need to do is check the latter.

@OliverHouse: But again: what you want is "preserve the structure". That's what makes it a category. Instead, you seem to be asking "what intrinsic property can I give a map to ensure that it preserves whatever 'the structure' happens to be?" but that is not a categorical question: that is precisely an "individual structure" property, which is exactly what category theory doesn't care about.

"Preserve relations":if $R$ corresponds to $R'$, and $aRb$ holds in the domain, then $f(a)Rf(b)$ should hold in the image. "Collections of subsets" is unclear, and depends on what you want to do with that collection of subspaces; sometimes you want direct images, sometimes you want inverse images. Operations are easy; partial operations as well: for binary, if $ab$ is defined, you want $f(a)*f(b)$ to be defined and equal $f(ab)$.

@ArturoMagidin I see what you mean about it this question not being (strictly speaking) about categories, but my motivation came from wanting to form concrete categories by choosing the most appropriate morphisms for the given objects. 'Preserve the structure' is vital, yes, but there is still the question of what is the most 'natural' choice. Usually when I come across a new concrete category and I know the objects and the morphisms are not given, I can guess what they are and am correct.

@ArturoMagidin By 'collections of subsets', I didn't just mean any, I mean ones that are meaningfully defined like topologies or $\sigma$-algebras. Given the objects, there are often many possible choices of morphisms to make it a category, some more sensible than others.

You can guess because you want them to "preserve the structure". Again: if you know what "the structure" is that you want to preserve, then the morphisms need to "preserve the structure" and it is clear what you want them to do. In most of mathematics, if there is something obvious to try, it almost always works (I can think of fewer than a handful of situation sin which this doesn't happen). "There are often many possible choices of morphisms": usually that's because there are many possible choices of structure to focus on.

I know that you "collection of subsets" can mean many things. But depending on what structure you are trying to preserve, what you do with those collections is different. In some situations, you want the direct image to be perserved (if $A$ is a distinguished subset of the domain, then $f(A)$ should be a distinguished subset of the codomain). In others, depending on the type of structure you are looking at, you want the inverse image to be preserved (if $A$ is a distinguished subset of the codomain, you want $f^{-1}(A)$ to be a subset of the codomain). Context.

Again: there are some obvious things to set; which ones you pick depends on which type of structure you are trying to preserve. Either your question has no possible answer because you are not clear what it is you are asking, or else the answer is already provided in the question itself.

@ArturoMagidin I explicitly acknowledged in the final paragraph of my question that there may be no definite answer to it. This is why I tagged it as a 'soft question', with the intention of shedding some light on it. However, the 'structure to focus on' is usually clear in many cases, such as with groups and topological spaces, is it not?

Which means that the choice of morphisms is clear, which means you aren't actually asking a question. Again: if you know what structure you want to preserve, then it is usually clear what you want the morphisms to preserve. When you have "multiple choices", it's usually because you actually have multiple structures (in your last example, to repeat, you could be focusing on distance or on topology, which is why you get the "choice" between isometries and continuous maps). So again: is your question "if the choice is obvious, how do you make the obvious choice?"

If someone told you the definition of a topology and you had no experience with continuous functions, you might predict that the morphisms should handle open sets well. How would you know to choose functions such that the preimage of an open set is open, as compared to those for which the image of an open set is open? Can you articulate why one of those "obviously" right?

@ArturoMagidin you could give an answer explaining your position (which I could then downvote) instead of leaving condescending comments...

@JohnPalmieri Good point! I guess without any prior knowledge of the significance of continuous functions I might have gone for the 'open maps' option for the morphisms. But, in the context of trying to generalise continuous functions on $\mathbb{R}$ to topological spaces, the 'preimages' one seems to be the clear choice

Note that the category of topological spaces with open maps is a perfectly fine category. It has its uses. Is it "wrong"? Not really; it's just not the one that captures the ideas of deforming/continuity.

@Arturo Magidin (and others): Regarding the note beginning with "That situation is related to this", the introductory portion of the following paper will be of interest: David Barry Gauld, Nearness − a better approach to topology, Mathematical Chronicle [after 1991: New Zealand Journal of Mathematics] 7 #1−2 (June 1978), pp. 84−90. MR 58 #12877; Zbl 387.54011 See also Applications of near sets.

@ArturoMagidin: I understand this, and the point I wanted convey to the original poster was that you need some sort of experience with the purported category (e.g., to go from nearness to open sets, or to go from metric spaces to topological spaces) before understanding what the morphisms should be. The choice of morphisms is not some abstract thing or something that can be done in isolation, it relies on the context and what you're trying to capture in defining the category.