Certainly Homomorphism is a prerequisite to establish an “Isomorphism”(Bijection), but what does a Homomorphism tell independently when it is established between two sets?
Homomorphism relates two sets as it is defined. But does it tell anything else? Or it is a tool for relating two sets only.
It would be nice to have an example where Homomorophism plays a big role besides being a condition for Isomorphism?
You are given two sets $A$ and $B$, both provided with a binary operation $*\>$. This means that in $A$ as well as in $B$ for certain triples $x$, $y$, $z$ it is true that $z=x*y\>$; e.g., $13=5+8$, or $91=7\cdot 13$. A map $\phi:\>A\to B$ is a homomorphism if it preserves such "incidences": $$z=x*y\quad\Longrightarrow\quad \phi(z)=\phi(x)*\phi(y)\ .$$