Consider a class of sets sharing some structure and properties e.g. groups, vector spaces, metric spaces, topological spaces, rings, fields. I am curious about under what conditions there may be a 'natural' choice of morphism that preserves this structure, such that the corresponding notion of 'isomorphism' allows us to consider two of them to be structurally identical.
My motivation for asking this question is to consider what the most natural choice of morphisms would be to form a concrete category, based on what is known about the structure and properties of what I would like to have as the objects.
I'll give some elementary examples:
Sets have no internal structure, so in a sense any function $f:A\to B$ is compatible with the (non-existent) structure on $A$ and $B$ and the morphisms are just functions. The only remotely salient feature of a set is its cardinality, which is preserved up to 'isomorphism' simply by bijections. Taking objects to be sets and morphisms to be functions between sets then forms the concrete category $\textbf{Set}$.
For groups, we attach a binary operation with various properties, so the morphisms $\varphi:G\to H$ would be ones that ensure that combining elements in $G$ corresponds with combining elements in $H$ (which then automatically preserve identities and inverses). I can't think what else one could possibly choose other than the standard choice: group homomorphisms, which gives the category $\textbf{Grp}$.
For topological spaces, we have a notion of what it means for a subset to be 'open', and continuous functions seem like the obvious choice: a continuous bijection with continuous inverse (a homeomorphism) between two topological spaces $X$ and $Y$ ensures that what it means for subsets to be 'open' corresponds perfectly between the two, so we consider them identical. Topological spaces together with continuous functions define the category $\textbf{Top}$.
Here's a more ambiguous one that makes me doubt whether there is a natural choice of morphism in many cases: normed vector spaces (say, over $\mathbb{R}$). We could form a concrete category using either bounded linear maps, or restrict to just isometric ones. We could even choose to use those linear maps with operator norm no greater than $1$. I'm guessing which of these would be most appropriate to use depends to a great extent on the context.
I am aware that there is probably no definitive answer to this, so I just hope that someone can shed some light on the issue. Given relations, collections of subsets etc., how would you go about finding the most appropriate choice of morphisms?