Relationship structure homomorphisms vs. category morphisms

Question

Different types of homomorphisms (e. g. group hom.s, vector space hom.s, algebra hom.s) seem to be able to be generalized using two strategies.

1. Either describe the homomorphism class as a class preserving a specific structure (structure-preserving hom.s, the ,,logician's way").
2. Or see them as morphisms within a category (the ,,category theorist's" way).

Are these two approaches always interchangeable or is one more potent than the other? Or do they have each their one expressive abilities not shared by the other one?

I searched SE and the Internet, but did not find similar questions.

Answer 1

If it's closed under composition, has associative composition, and has identities, it's already a category. Whether you remark on that fact by calling it a category is up to you, along with whether or not you feel like using a theorem proven about categories to do whatever it is you're doing. It's kind of like how real-valued functions are vectors whether you call them vectors or not.

Answer 2

There’s no dichotomy between these viewpoints. They’re compatible, and are both very important and useful. At this point, they’ve been used together so much that it’s not really possible to separate their achievements and ask which one of them is “more potent”.

From the point of view of logic — specifically, universal algebra — categories give you a very useful organising tool for your structures and homomorphisms, and lots of general theorems that apply to them. From the point of view of category theory, categories of “algebraic structures and their homomorphisms” form a very important class of categories. Indeed, they give several different such classes of categories, depending on what kind of “algebraic structures” you consider. They’re very well studied, under terms like locally finitely presentable categories, and they enjoy many nice and useful properties.