Outside of the technical definitions, what exactly is a homormorphism or an isomorphism "saying"?
For instance, let's we have a group or ring homomorphism $f$, from $A$ to $B$. Does a homomorphism mean that $f$ can send some $a_i$ in $A$ to $b_j$ in $B$, but has no way to "get it back"?
Similarly, if we have a group or ring isomorphism $g$ from $A$ to $B$, does it mean that $g$ can both send and "take back" some $a_i$ in $A$ to/from $b_j$ in $B$?
I'm sorry if this question sounds stupid, but I'm just trying to understand the meaning behind homomorphisms and isomorphisms outside of the technical definitions. I think it will help me tremendously to be able to put them into "dumbed down" definitions. Thank you for your help!