I have a general questions about category theory (I don't know many things about this area, so please, be indulgent if these questions are a bit fuzzy)
1) If we consider for example the class of groups and two groups $(G, +)$ and $(H, ×)$. We will say that $f : G \rightarrow H$ is a morphism if : $\forall (x, y) \in G \times G, f(x + y) = f(x) × f(y)$, right ? But why is that so ? I mean, why this definition of morphism between groups is admitted and not another one ?
2) If we want to consider a new kind of class, how can we find a "natural" definition of morphism between two structures of this class ?
Thank you for your help !