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?

On a general note: A mathematical object, presented in some way, is well-defined if it really is what it’s implicitly or explicity claimed to be in its presentation. So saying “The map $φ \colon G → G'$ is a well-defined group homomorphism,” means “The map $φ \colon G → G'$ really is a map and, moreover, a group homomorphism.”
Proving well-definedness of a mathematical object is therefore appropriate whenever there’s reasonable doubt of it, as José Carlos Santos has already pointed out.
So bottom line: You have to apply common sense.
One way to do this is the following: Play devil’s advocate and try to find an ill-defined object in your writing, being very nitpicky about it. If you spot any chance of exposing something to be ill-defined and you tackle it, but it withstands and really turns out to be well-defined, but you had to convince yourself first (perhaps even by writing down an argument) – that’s when you should give the argument to prove the object is well-defined.
