When is it necessary to prove well definedness? Last semester, we tackled group theory. Our professor did not require us to show that a mapping $\phi:G\to G'$ is well-defined before proving that it is a group homomorphism. Now, our new professor requires us to show that $\phi$ is well-defined first before proving that $\phi:R\to R'$ is a ring homomorphism. So, in general, when is it necessary to show that a mapping is well-defined?

When there's a real chance that the map is not well-defined. For instance, the map $\psi\colon\mathbb{Z}_2\longrightarrow\mathbb Z$ defined by $\psi(\overline x)=x$ is not well-defined, since $\overline 0=\overline 2$ but $\psi(\overline0)\neq\psi(\overline2)$.
