Is there a unifying notion of "structure" that can be referenced when saying that a morphism is structure preserving? I have read the following:

*

*Why is there apparently no general notion of structure-homomorphism?

*What does Structure-Preserving mean?

*Can we deduce that morphisms in categories of structures should be “structure preserving”

*Morphisms has to be structure preserving?

*Is every abstract category a concrete category of structures?
They all seem to be attempting to scratch the same itch, and I'm unsatisfied with the answers. It seems to me that in order for someone to answer questions like "do morphisms have to be structure preserving?" we would first need to agree on what structure is.
Question
Is structure in the sense that we might say "FOO is a structure preserving morphism in the category of BAR" formalized? For example, is there consensus on a definition that generalizes the structure preserved by continuous maps in the Top category, the structure preserved by group homomorphisms in the Grp category, and the structure preserved by smooth maps in the Man category?
The wiki article for mathematical structure seems a bit vague.
 A: An "algebraic structure" can be defined as a set together with a collection of formal logical sentences satisfied by the set. These sentences are usually called the axioms of the structure, and structure-preserving maps may be defined as those such that the sentences remain true when applied to the image set of the morphism.
I'm sure you've read about examples, such as the thin category's generalization of the poset, wherein morphisms are not structure-preserving. Such things' ubiquity is why the idea that morphisms are "structure-preserving" is little more than a pedagogical aid; the question of how morphisms act on structure is perhaps the definition of standard abstract algebra.
Working directly with such a definition of "algebraic structure" according to such a definition has a storied history; see for example universal algebra. The basic idea of category theory is to understand such algebraic structure without looking at structures themselves, only the maps between classes of structures. In my opinion, category theory is by far the best way to do such meta-analysis. It actually has predictive power, allowing one to relate wildly distant algebraic structures to one another (an example I'm familiar with is categorical quantum mechanics connecting Hilb to linear logics), whereas other approaches feel more ad-hoc. Furthermore, it is truly universal: one of the first categories one reads about is Cat, which describes the structure of category theory in terms of itself. Furthermore, some current researchers believe categories may end up being used to describe all of mathematics, even formal logic.
TL;DR: while one can can rigorously define structure, the (informal) idea behind category theory is that you don't need to.
