TITLE: What does Structure-Preserving mean?
QUESTION [14 upvotes]: A very basic definition in category theory is the definition of morphism between objects.
If the category is a construct, i.e., a category $\mathcal C$ equipped with a faithful functor $U\colon \mathcal C\to Set$, how can the morphisms be seen as structure-preserving?
In other words, what exactly does "structure-preserving" mean? If I have two topological spaces, why are continuous functions "structure-preserving" functions?
Has it something to do with the "final topology" or "initial topology" of a continuous function?

REPLY [5 votes]: Good question.
In algebra, "structure-preserving" is easy to define, because algebraic structures can be viewed as product-preserving functors out of Lawvere theories, and the homomorphisms between them are just the natural transformations.
However outside of algebra, it becomes much trickier to work out what is the "correct" notion of homomorphism, and sometimes there is more than one "correct" answer, depending on what you're trying to do.