In “Arbib, Manes. Arrows, Structures and Functors. The Categorical Imperative. 6. Structured sets.” there is an approach to formalize structures. I have a strong feeling that they describe just concrete categories in other words. For a concrete category $(C, U)$, the **set of structures on a set** $X$ (this is Arbib, Manes's term) is $(fr\\_ob(U))^{-1}(X)$ where $fr\\_ob(U)$ is a mapping of $U$ on objects. Similarly, the set of **admissible functions** $A\to B$ (for $A\ B\in ob(C)$) is $im(fr\\_hom(U)(A, B))$ where $fr\\_hom(U)$ is a mapping of $U$ on morphisms.

If I am on the right track, what do other parts of Arbib, Manes's theory (e.g. optimal lifts) correspond to?

(Please add the tag “concrete-category” to this question.)