Is there a bijection between kinds of mathematical structures and morphisms that preserve all properties of this kind of structure?
Topologycal spaces $\leftrightarrow$ Homeomorphisms
Metric spaces $\leftrightarrow$ Isometries
Sets $\leftrightarrow$ Bijections
Groups $\leftrightarrow$ Isomorphisms of groups
Rings $\leftrightarrow$ Isomorphisms of rings
Modules $\leftrightarrow$ Isomorphisms of moodules
Being more precise, my question becomes in 2 particular questions in concrete categories:
Given a mathematical structure in a set (concrete object), ¿this follows which functions are the morphisms?.
Given a function between sets, ¿this follows which objects must be the sets?.